text:
Cinnamon Network International
Business Leaders Offer Free Clinics to Emerging Entrepreneurs
Budding entrepreneurs can take advantage of free coaching from world-class advisers as part of a campaign to mobilise the Church and spearhead economic growth.
Cinnamon Network International's Spirit of Enterprise campaign will be facilitating monthly online clinics, with the first two already scheduled for March and April. These clinics will give people with entrepreneurial ideas the inside track on setting up and running their own businesses.
Business Coach Lisa Chee MBE will kick-off the on-line coaching clinics by hosting 'Nail Your Niche' on Wednesday 31 March 2021.
Lisa, a 'foodie' entrepreneur, was awarded an MBE for selling her range of sauces to the Queen. She now runs Start Up Coach supporting others on their entrepreneurship journey and has co-authored 3 best selling cookbooks.
Next month the coach to share their experience will be Bishop Wayne Malcolm – well known as 'The Business Bishop'.
A leading performance coach who has produced numerous learning programmes, Bishop Malcolm will help delegates understand how to spot and make the most of opportunities.
Lisa Chee, who has created and built several six-figure turnover businesses, comments: "A recent report showed that 80 per cent of businesses across the UK fail within their first year.
According to the latest figures from the Office for National Statistics, only 42.4 per cent of businesses started in 2013 were still trading five years later.
Part of the problem is that emerging entrepreneurs don't get the right advice from the start, so can't spot issues until it's too late. The Spirit of Enterprise campaign is all about faith-driven entrepreneurs helping the Church support people who have ideas to start a transformational business.
To do that effectively, we need to offer the best advice from those who have extensive experience in running a business or businesses, " added Lisa, who is a member of the Spirit of Enterprise steering group."
She said if entrepreneurs could access experienced coaches and fill their skills gaps they would be more likely to succeed.
Her 'Nail Your Niche' clinic will help people use their skills and experience to find and develop the niche for their business. The one-hour session starts at 1pm on 31 March and will take place online. To book a place, visit the Spirit of Enterprise website, https://www.thespiritofenterprise.net/clinics
The Spirit of Enterprise is a global partnership encouraging the God-given gift of enterprise by supporting people with ideas to start a transformational business.
The founder of the Spirit of Enterprise, Matt Bird, says: "Church food banks are doing an amazing job in light of the pandemic, and while they can feed people for a few days, helping people start a business could help feed them for life.
Poverty can be relieved by charity, but the only sustainable solution to poverty is to create jobs and ethical business. In a crisis situation emergency relief deals with immediate needs but strategic development builds a new and enduring future."
The not-for-profit global partnership, facilitated by Cinnamon Network International, includes organisations such as Faith Driven Entrepreneur, the Transformational Enterprise Network, The Lions, and Christian Aid's business network, Salt.
Cinnamon's aim is to help transform communities, cities and nations, through identifying opportunities that create societal value, sustainable employment and economic growth.
London, SW19 1YW, UK
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The house is located in Alpine Mountain Ranch & Club. This photo was taken near the property. The spectacular surroundings and regional style inspired the architecture of the home.
The wings and rooflines break up the large home into smaller volumes of pleasing proportions. The two-story section containing the public spaces is in the middle, the front entrance is to the left, and a three-car garage is to the right. The facade mixes metal cladding, stone, wood and big expanses of glass. The glass is triple-paned to keep the house sealed against the cold winter temperatures.
Views of the mountains draw people from the entry into the main living space. White oak brings the rustic warmth of wood grain to the floors. "Using lots of wood added a cozy feel to the slick house," Dapper says. The chandelier-lit staircase leads to four bedrooms on the walk-out lower level.
The kitchen sits in the heart of the main level's open floor plan. A two-sided fireplace provides warmth and some separation between this space and the living room. Wood beams, steel trusses, clerestory windows and a large light fixture bring the soaring vaulted ceiling down to human scale.
"Figuring out how to provide enough task lighting with the high ceilings was a challenge," Dapper says. "But it was a chance to fill up the space and make it feel cozier. And we can do so much with LED technology these days." Creating the custom light fixture was a big effort. Dapper used SketchUp to design it so that she could create 3D images and rotate it around to see how it looked. Then she laid plywood templates on the kitchen floor to get the scale right. Local company Fedewa Custom Works fabricated the fixture.
"The kitchen's U is finished in high gloss and is shiny to the point that it almost shimmers," Dapper says. "We mixed in the wood laminate finish on the appliance wall to create a rustic and slick combination in here." The light fixture also has a high-gloss finish to tie it to the cabinetry below.
Dapper also designed an LED light rail to illuminate the appliance wall from above.
The countertops are Caesarstone quartz in Blizzard White. Their thickness suits the scale of the room, while waterfall countertops add to the sleekness of the space. The view outside the windows is of the Steamboat Ski Resort.
The dining area is on the other side of the kitchen, separated by the U-shaped island. The side of the island facing the dining room can serve as a buffet.
The linear light fixture plays off the strong lines seen throughout the space and visually lowers the ceiling.
Even with the high vaulted ceilings and large expanses of glass, the house stays warm, Dapper says, thanks to triple-paned glass, in-floor heating, the fireplace and a backup forced hot air system.
"This is a great party house," Dapper says. The TV surround and fireplace block the views of the kitchen work zone while leaving them open to the ceiling, light fixture and ski resort views on the opposite side of the house. The fireplace surround is rolled steel (modern), and the hearth and TV surround are travertine (rustic). Large sofas and chairs suit the scale of the room and are cushy for lounging. A dark rug softens the space and picks up the gray-blue tones in the fireplace surround.
Large sliding doors open to a deck that serves guests well in warmer months, when fishing and hiking replace skiing and snowshoeing as the top outdoor activities.
The master suite is on the main level in the wing seen on the right side of the previous photo. The decor takes a backseat to the stunning surroundings — the framed views serve as artwork.
"We put the wood on the wall instead of the floors in here," Dapper says. This accent wall balances the modern windows and contemporary light fixture in the mountain-modern mix, while dark trim and simple, Scandinavian-inspired bedding warm up the space. The room also has direct deck access to another deck through the large sliding glass doors.
Two chairs in front of the picture window provide a spot to enjoy the views and brace for the outdoor adventures the day will bring.
The master bathroom maintains an open and expansive feeling. Long horizontal lines and mirrors with integrated lighting maintain a minimalist look, warmed by wood finishes and wood-look porcelain tile. The countertops are Taj Mahal quartzite.
Clear glass on the shower enclosure keeps views through the room open from the sculptural bathtub. Wood-colored tile with a ribbed texture adds warmth and interest around the bathtub.
The entry and kitchen seem sterile to me - the house didn't "warm up" until the photos got to the living room and bedrooms. Gorgeous architecture and space, just needs more texture IMHO.


text:
\section{Introduction}\label{intro}
Throughout the article, all topological spaces are assumed to be Tychonoff.
By $C_{p}(X)$ we mean the space of all real-valued continuous functions on a Tychonoff space $X$
 endowed with the topology of pointwise convergence.

\begin{deff}{\rm (\cite{KL}, \cite{Knight})}\label{def:Delta}
A topological space $X$ is said to be a $\Delta$-space if for every decreasing sequence $\{D_n: n \in \omega\}$
of subsets of $X$ with empty intersection, there is a decreasing sequence $\{V_n: n \in \omega\}$ consisting of open
subsets of $X$, also with empty intersection, and such that $D_n \subset V_n$ for every $n \in \omega$.
\end{deff}

The class of all $\Delta$-spaces is denoted by $\Delta$.
Let us point out that the original definition of a $\Delta$-set $X \subset \R$, where $\R$ denotes the real line,
is due to G. M. Reed and E. K. van Douwen (see \cite{Reed}).
$\Delta$-sets of reals have been used and investigated thoroughly in the study of two of the most basic and central constructions in general topology:
the Moore--Nemytskii plane and the Pixley-Roy topology.
Denote by $M(X)$ the subspace of the Moore--Nemytskii plane, which is obtained by using only a subset $X \subset \R$ of the $x$-axis.
G. M. Reed observed that $M(X)$ is countably paracompact if and only if $X$ is a $\Delta$-set \cite{Reed}.

For a separable metrizable space $M$, denote by $\fcal(M)$ the hyperspace of finite subsets of $M$ endowed with the Pixley-Roy topology.
D. J. Lutzer proved that if $M$ is a strong $\Delta$-set, i.e. every finite power $M^n$ is a $\Delta$-set, then $\fcal(M)$ is countably paracompact \cite{Lutzer}.
H. Tanaka proved the converse statement: if $\fcal(M)$ is countably paracompact, then $M$ is a strong $\Delta$-set \cite{Tanaka}.
Also, the work \cite{Tanaka} deals with the analogous questions for general (not necessarily separable) metrizable spaces.

A set of reals $X$ is called a $Q$-set if each subset of $X$ is $F_{\sigma}$, or, equivalently,
each subset of $X$ is $G_{\delta}$ in $X$. The existence of uncountable $Q$-sets is independent of ZFC. Every $Q$-set is a $\Delta$-set, but consistently the converse is not true
(see\cite{Knight}). More details about $Q$-sets and $\Delta$-sets can be found in \cite{FM}, \cite{Knight}.
Of course, there are plenty of nonmetrizable $\Delta$-spaces with non-$G_{\delta}$ subsets, in ZFC \cite{KL}.

We could not find a single paper devoted to investigation of the general topological $\Delta$-spaces.
Quite recently the authors have shown that the notion of $\Delta$-spaces plays a key role in the study of distinguished $C_p$-spaces \cite{KL}.
\begin{theorem}\cite{KL}\label{Theor:description}
For a Tychonoff space $X$, the following conditions are equivalent{\rm:}
\begin{enumerate}
\item[{\rm (1)}] $C_p(X)$ is distinguished.
\item[{\rm (2)}] Any countable disjoint collection of subsets of $X$ admits a point-finite open expansion in $X$.
\item[{\rm (3)}] $X$ is a $\Delta$-space.
\end{enumerate}
\end{theorem}
We should mention that independently and simultaneously an analogous description of distinguished $C_p$-spaces (but formulated in different terms)
appeared in \cite{FS}.

By a \emph{bounded set} in a topological vector space  we understand any set which is absorbed by every $0$-neighbourhood.
Following  J. Dieudonn\'{e} and
L. Schwartz  \cite{dieudonne} a locally convex space (lcs) $E$ is called \emph{distinguished} if every bounded subset of the bidual of $E$
 in the weak$^{*}$-topology is contained in the closure of the weak$^{*}$-topology  of  some bounded subset of $E$.
Equivalently, a lcs $E$ is distinguished if and only if the strong dual of $E$  (i.e. the topological  dual of $E$ endowed with the strong topology)
is \emph{barrelled}. A. Grothendieck  \cite{grothendieck} proved that a metrizable lcs $E$ is distinguished
if and only if its strong dual is \emph{bornological}.
Recall that the \emph{strong topology} on $E^{\prime}$ is the topology of uniform convergence on bounded subsets of $E$.

Denote by $L_p(X)$ the dual of $C_p(X)$, i.e. the linear space of all continuous linear functionals on $C_p(X)$, endowed with the topology of pointwise convergence.
Basic properties of $L_p(X)$ are described thoroughly in \cite{Arch5}.
By $L_{\beta}(X)$ we denote the strong dual of $C_{p}(X)$, i.e. the space $L_p(X)$ endowed with the \emph{ strong topology} $\beta_{X}=\beta(L_p(X), C_{p}(X))$.
 Note also that for a vector space $E$ the finest locally convex topology $\xi$ of $E$ is generated by the family of  all absolutely convex and absorbing subsets of $E$
which form a base of neighbourhoods of zero for the topology $\xi$.

The following main characterization theorem has been proved instrumental in the study of distinguished lcs $C_p(X)$.\\
\\

\begin{theorem}[\cite{FKLS}, \cite{fe-ka}]\label{three}
For a Tychonoff space $X$ the following assertions are equivalent:
\begin{enumerate}
\item[{\rm (1)}] $C_{p}(X)$ is distinguished, i.e. its strong dual $L_{\beta}(X)$ is a barrelled space.
\item[{\rm (2)}] For each $f\in\mathbb{R}^{X}$ there is a bounded  $B\subset C_{p}(X)$ with $f\in \cl_{\R^X}(B)$.
\item[{\rm (3)}] The strong topology $\beta_{X}$ of the strong dual of $C_{p}(X)$ is the finest locally convex topology on $L_{p}(X)$.
\end{enumerate}
\end{theorem}

Naturally, aforementioned crucial Theorem \ref{Theor:description} has been proved in \cite{KL} with the help of Theorem \ref{three}.
In this paper, Theorem \ref{three} has been applied effectively again for the proof of Theorem \ref{linear1}, the main result of Section \ref{Linear}.

Our aim is to continue the research about topological $\Delta$-spaces originated in our paper \cite{KL}.
We obtain results in two directions.
First, in Section \ref{Images} we investigate whether the class $\Delta$ is invariant under the basic topological operations, including
continuous images, closed continuous images, countable unions and finite products.
\emph{What do we know about continuous images of $\Delta$-spaces?}

\begin{proposition}\cite{KL} \label{prop:locomp}
There exists in ZFC a MAD family $\acal$ on $\N$ such that the corresponding Isbell--Mr\'owka
space $\Psi(\acal)$ admits a continuous mapping onto the closed interval $[0,1]$.
\end{proposition}
Thus, the class $\Delta$ is not invariant under continuous images even for first-countable separable locally compact pseudocompact spaces.
The following result has been proved in our paper \cite{KL}.
\begin{proposition}\cite{KL} \label{prop:closedmap} Let $X$ be any $\Delta$-space and $\varphi: X \to Y$ be a closed continuous surjection with finite fibers.
Then $Y$ is also a $\Delta$-space.
\end{proposition}

Shortly after the paper \cite{KL} was published, V. Tkachuk \cite{Tkachuk} observed that the proof of Proposition \ref{prop:closedmap} in fact
does not use the last restriction about finiteness of fibers.
 So, Proposition \ref{prop:closedmap} is valid without unnecessary assumption of finiteness of fibers and the class $\Delta$ is invariant under closed continuous images.
As an immediate consequence, V. Tkachuk \cite{Tkachuk} noticed that we have a positive answer to Problem 5.3 posed in \cite{KL}:
any continuous image of a compact $\Delta$-space is also a $\Delta$-space.

In this paper we generalize Proposition \ref{prop:closedmap} as follows:
 Let $X$ be any $\Delta$-space and $\varphi: X \to Y$ be a continuous surjection
such that $\varphi(F)$ is an $F_{\sigma}$-set in $Y$ for every closed set $F \subset X$;
then $Y$ is also a $\Delta$-space (Theorem \ref{th:F-sigma_map}).
 It is interesting to note that the proof of Theorem \ref{th:F-sigma_map}
is obtained by absolutely elementary arguments.

We say that a topological space $X$ is \emph{$\sigma$-closed discrete} if $X = \bigcup_{n\in\omega}X_n$, where each $X_n$ is a closed and discrete subset of $X$.
It is easy to see that every $\sigma$-closed discrete space is in $\Delta$.
A straightforward application of Theorem \ref{th:F-sigma_map} gives a far-reaching generalization of this fact:
Assume that $X$ is a countable union of closed subsets $X_n$, where each $X_n\in \Delta$; then also $X\in \Delta$ (Proposition \ref{prop:count_unions}).
As a corollary we solve in the affirmative Problem 5.8 posed in \cite{KL}:
a countable union of compact $\Delta$-spaces is also a $\Delta$-space.
In particular, $\sigma$-product of any family consisting of scattered Eberlein compact spaces is a $\Delta$-space.
Another consequence says that the product of a $\Delta$-space with a $\sigma$-closed discrete space
(in particular, a countable space) is a $\Delta$-space.
Remark that the general question whether the class $\Delta$ is invariant under finite products remains open.
It is worthwhile mentioning that we do need an assumption on finite fibers for the following "reverse" version of Proposition \ref{prop:closedmap} above:
Let $\varphi: X \to Y$ be a continuous surjection with finite fibers; then $Y \in \Delta$ implies that also $X \in \Delta$ (Proposition \ref{prop:reverse}).

Following A. V. Arkhangel'skii \cite{Arch4}, we say that a space $Y$ is $l$-dominated ($u$-dominated, $t$-dominated)
by a space $X$ if $C_p(X)$ can be mapped linearly and continuously (uniformly continuously, continuously, respectively) onto $C_p(Y)$.
There are many topological properties which are invariant under defined above relations, and there are many which are not.
The main goal of Section \ref{Linear} is to study the following question: \emph{Which topological properties related to being a $\Delta$-space are preserved by
the relation of $l$-dominance}?

We show that the class of Tychonoff spaces $\Delta$ is invariant under the relation of $l$-dominance, equivalently, the class of distinguished $C_p$-spaces
is invariant under the operation of taking continuous linear images (Theorem \ref{linear1}).
For the reader's benefit, aiming to emphasize a big potential in this research area,
we present two different proofs of Theorem \ref{linear1}. The first proof is based on item (2) of Theorem \ref{three} and invokes a new observation
about extensions of linear continuous surjections between $C_p$-spaces. The second proof uses the language of dualities and is based on item (3) of Theorem \ref{three}.
As an easy consequence of our approach, we obtain that inside the class of subsets of $\R$, $Q$-sets are invariant under the relation of $l$-dominance (Corollary \ref{Q-set}).
What makes the proof of Corollary \ref{Q-set} surprising is the fact that it does not depend on the question whether the square of a $Q$-set remains a $Q$-set.
We prove also that such topological properties as $\sigma$-scattered, $\sigma$-discrete, Eberlein compact, scattered Eberlein compact are preserved by the relation of $l$-dominance.

The last Section \ref{PFA} is devoted to the study of compact / countably compact $\Delta$-spaces.
It has been shown in \cite{LS} that every compact $\Delta$-space has countable tightness.
Relying on Proposition \ref{prop:closedmap}, we  provide a very short proof of this assertion.
As a consequence, under  Proper Forcing Axiom (PFA) every compact $\Delta$-space is sequential.
In \cite{KL} we proved that every compact $\Delta$-space $X$ is \emph{scattered}, i.e. every subset $A$ of $X$ has an isolated (in $A$) point.
Making use of Proposition \ref{prop:closedmap} again, in Theorem \ref{th:count_compact} we generalize this result for countably compact $\Delta$-spaces, in ZFC.

In order to better understand the boundaries of the class $\Delta$ one should search for new examples and counter-examples.
Another recent relevant paper \cite{LS} is devoted to the following question: under what conditions 1) trees topologies;
2) $\Psi$-spaces built on maximal almost disjoint families of countable sets; 3) ladder system spaces do belong to the class $\Delta$.
Note that there are compact scattered spaces $X \notin \Delta$ (for example, the compact space $[0,\omega_{1}]$) \cite{KL}.
A stronger result has been obtained in \cite{LS}: there exists a compact scattered space $X$ such that the scattered height of $X$ is finite, and yet $X \notin \Delta$.
Thus, Problem 5.11 from \cite{KL} has been solved negatively in \cite{LS}.

Our notations are standard, the reader is advised to consult with the monographs \cite{Arch5} and \cite{Engelking} for the notions which are not explicitly defined in the text.
In the article we pose a dozen open questions.
\section{Continuous images, unions and products of $\Delta$-spaces}\label{Images}

\begin{theorem}\label{th:F-sigma_map} Let $X$ be a $\Delta$-space and $\varphi: X \to Y$ be a continuous surjection
such that $\varphi(F)$ is an $F_{\sigma}$-set in $Y$ for every closed set $F \subset X$.
Then $Y$ is also a $\Delta$-space.
\end{theorem}
\begin{proof}
Assume that $\{D_n: n \in \omega\}$ is any decreasing sequence of subsets of $Y$ with empty intersection.
 Then $\{\varphi^{-1}(D_n): n \in \omega\}$ is a decreasing sequence of subsets of $X$ and
$\bigcap_{n\in\omega}\varphi^{-1}(D_n) = \varphi^{-1}(\bigcap_{n\in\omega}D_n) = \emptyset$.
By assumption, there is a decreasing sequence of open sets $\{U_n: n \in \omega\}$ such that $\varphi^{-1}(D_n) \subset U_n$ for each $n \in \omega$ and $\bigcap_{n\in\omega} U_n = \emptyset$.
Define $H_n = Y\setminus \varphi(X\setminus U_n)$ for each $n\in\omega$.
We have that $D_n \subset H_n$ for each $n\in\omega$. Indeed, $\varphi^{-1}(y) \subset U_n$ for every $y \in D_n$, hence
$y \notin \varphi(X\setminus U_n)$ which means that $y \in H_n$.
Note that $\varphi^{-1}(H_n) \subset U_n$, therefore $\bigcap_{n\in\omega} H_n = \emptyset$
 because $$\varphi^{-1}(\bigcap_{n\in\omega} H_n) = \bigcap_{n\in\omega} \varphi^{-1}(H_n) \subset  \bigcap_{n\in\omega} U_n = \emptyset$$
Clearly, the sets  $X\setminus U_n$ are closed, the sets $\varphi(X\setminus U_n)$ are $F_{\sigma}$ in $Y$, hence the sets $H_n$ are $G_{\delta}$ in $Y$.

It appears that we can refine the sets $H_n$ further and construct another decreasing sequence consisting of open sets $\{V_n: n \in \omega\}$ such that $H_n \subset V_n$ for each $n \in \omega$ and
 $\bigcap_{n\in\omega} V_n = \emptyset$. This claim in fact should be attributed to E. K. van Douwen (see \cite[page 150]{Reed}).
Because the real argument has not been provided in \cite{Reed}, for the sake of completeness we include the proof. Denote  by $H_n^k$ open subsets of $Y$ such that $H_{n+1}^k \subset H_n^k$
and $\bigcap_{n\in\omega}H_n^k = H_k$ for every $n, k \in \omega$. Since the sequence $\{H_k: k \in \omega\}$ is decreasing, by induction over upper index $k$ without loss of generality
we may assume also that $H_{n}^{k+1} \subset H_n^k$ for every $n, k \in \omega$. We declare now that $V_n = H_n^n$, $n \in \omega$. It is clear that $D_n \subset V_n$ and each $V_n$ is an open set.
We show that the intersection of all sets $V_n$ is empty. Indeed, let $y$ be any element of $Y$. There exists $k\in\omega$ such that $y \notin H_k$, hence there exists $n(k) \in \omega$ such that
$y \notin H_{n(k)}^k$. Fix any $m \geq \max\{k, n(k)\}$. Then $H_m^m \subset H_m^k \subset H_{n(k)}^k$ which implies that $y \notin H_m^m$. Finally,
$\bigcap_{n\in\omega}V_n = \emptyset$ and the proof is complete.
\end{proof}

\begin{proposition}\label{prop:count_unions}
Assume that $X$ is a countable union of closed subsets $X_n$, where each $X_n$ belongs to the class $\Delta$.
Then $X$ also belongs to $\Delta$. In particular, a countable union of compact $\Delta$-spaces is also a $\Delta$-space.
\end{proposition}
\begin{proof} Denote by $Z$ the free topological union of the spaces $X_n, n \in \omega$. It is easy to see that $Z \in \Delta$, by Theorem \ref{Theor:description}.
The space $Z$ admits a natural continuous mapping $\varphi$ onto $X$.
Since $\varphi(F)$ is an $F_{\sigma}$-set in $X$ for every closed set $F \subset Z$, we deduce that $X \in \Delta$, by Theorem \ref{th:F-sigma_map}.
 \end{proof}

Thus, we have a positive solution of Problem 5.8 posed in \cite{KL}.

\begin{corollary} \label{cor:cont_comp} Let $X$ be a $\sigma$-compact $\Delta$-space and $Y$ be a continuous image of $X$.
Then $Y$ also is a $\Delta$-space.
\end{corollary}

Thus, we have a positive solution of Problem 5.3 posed in \cite{KL}.

\begin{corollary}\label{cor:sigma_product}
$\sigma$-product of any family consisting of scattered Eberlein compact spaces is a $\Delta$-space.
\end{corollary}
\begin{proof} $\sigma$-product is a countable union of $\sigma_n$-products, where $\sigma_n$-product includes elements of the product
which support consists of at most $n$ points, $n \in \omega$ . Every $\sigma_n$-product of scattered Eberlein compact spaces is again a scattered Eberlein compact,
therefore it is a $\Delta$-space \cite{FKLS}. It remains to apply Proposition \ref{prop:count_unions}.
\end{proof}

\begin{corollary}\label{Lin} Let $X$ be a Lindel\" of subspace of a $\sigma$-product of any family consisting of scattered Eberlein compact spaces
and $Y$ be a continuous image of $X$. Then $Y$ also is a $\Delta$-space.
\end{corollary}
\begin{proof} $X$ is equal to the countable union of its closed subspaces $X_n$, where $X_n$ is the intersection of $X$ with $\sigma_n$-product.
Every Lindel\" of subspace of a scattered Eberlein compact is necessarily $\sigma$-compact, by a recent result of V. Tkachuk \cite{Tkachuk2}.
Finally, $Y$ is a $\dcal$-space by Corollary \ref{cor:cont_comp}.
\end{proof}

\begin{corollary}\label{Lin_Cech} Let $X$ be a Lindel\" of \v Cech-complete $\Delta$-space
and $Y$ be a continuous image of $X$. Then $Y$ also is a $\Delta$-space.
\end{corollary}
\begin{proof} Any Cech-complete $\Delta$-space is scattered \cite{KL}. Now we use the well-known fact stating that every Lindel\" of \v Cech-complete scattered
space is $\sigma$-compact (see \cite[Theorem 4.5]{Aviles}) and we finish the proof again by Corollary \ref{cor:cont_comp}.
\end{proof}

We don't know  answers to the following problems.

\begin{problem}\label{prob1}
Let $X$ be any Lindel\" of subspace of a compact $\Delta$-space. Is $X$ a $\sigma$-compact space?
\end{problem}

In the case that the answer to Problem \ref{prob1} is negative we can ask

\begin{problem}\label{prob2}
Let $X$ be any Lindel\" of subspace of a compact $\Delta$-space and $Y$ be a continuous image of $X$.
Is $Y$ a $\Delta$-space?
\end{problem}

We have a partial positive result for products of $\Delta$-spaces.

\begin{corollary}\label{product}
Let $Z$ be the product of a $\Delta$-space $X$ with with a $\sigma$-closed discrete space
(in particular, a countable space) $Y$. Then $Z$ also is a $\Delta$-space.
\end{corollary}
\begin{proof} Let $Y= \bigcup_{n\in\omega} Y_n$, where each $Y_n$ is a closed and discrete subset of $Y$.
Denote by $Z_n = X \times Y_n$. It is clear that each $Z_n$ is closed in $Z$ and $Z_n \in \Delta$.
We get that $Z$ is a countable union of closed $\Delta$-spaces $Z_n$, so Proposition \ref{prop:count_unions} applies.
\end{proof}

Next statement formally is more general than Proposition \ref{prop:count_unions}.

\begin{proposition}\label{prop:sigma-loc_finite}
Assume that $X$ is covered by a $\sigma$-locally finite family of closed subsets $\{X_{\alpha}: \alpha \in A\}$, where every $X_{\alpha}$ belongs to $\Delta$.
Then $X$ also belongs to $\Delta$.
\end{proposition}
\begin{proof} The union of a locally finite family of closed subsets of $X$ is closed in $X$ \cite{Engelking}.
Now remind the following fact \cite[Theorem 2.7]{Burke}. Suppose that $\pcal$ is a topological property preserved
under closed mappings and $\{X_{\alpha}: \alpha \in A\}$
is a locally finite closed cover of $X$ with each $X_{\alpha}$ satisfying $\pcal$. If the
free topological sum satisfies $\pcal$ then so does $X$. It suffices to say that $X$ satisfies property $\pcal$ if $X\in \Delta$ and apply Proposition \ref{prop:count_unions}.
\end{proof}

\begin{remark} Proposition \ref{prop:count_unions} is not valid without assuming that all pieces $X_n$ in the union are closed.
 Let $M$ be the Michael line which is the refinement of the real line $\R$ obtained by isolating all irrational points.
Clearly, $M$ can be represented as a countable disjoint union of singletons (rationals) and an open discrete set.
 Nevertheless, the Michael line $M$ is not in $\Delta$ \cite{FKLS}.
\end{remark}

Now we consider a question of "reversing" of Proposition \ref{prop:closedmap}.
It is evident that if $\varphi: X \to Y$ is a continuous one-to-one mapping from $X$ onto $Y$ and $Y$ is a $\Delta$-space, then $X$ is also a $\Delta$-space.
The following more general result has been conjectured by V. Tkachuk \cite{Tkachuk} and below we provide a straightforward argument.

\begin{proposition} \label{prop:reverse} Let $\varphi: X \to Y$ be a continuous finite-to-one surjective mapping.
If $Y$ is a $\Delta$-space, then $X$ is also a $\Delta$-space.
\end{proposition}
\begin{proof}
Let $\{B_n: n \in \omega\}$ be any decreasing sequence of subsets of $X$ with empty intersection.
Denote by $D_n = \varphi(B_n)$. Then $\{D_n: n \in \omega\}$ is a decreasing sequence of subsets of $Y$ with empty intersection.
Indeed, for every $y\in Y$ the fiber $\varphi^{-1}(y)$ is finite, hence there is $n \in \omega$
such that $\varphi^{-1}(y) \cap B_n = \emptyset$ which means that $y \notin D_n$.
By assumption, there is a decreasing sequence of open sets in $Y$,  $\{U_n: n \in \omega\}$ such that $D_n \subset U_n$ for each $n \in \omega$ and $\bigcap_{n\in\omega} U_n = \emptyset$.
Define $V_n = \varphi^{-1}(U_n)$ for each $n\in\omega$. Clearly, a decreasing sequence of open in $X$ sets, $\{U_n: n \in \omega\}$ is as required.
\end{proof}

We don't know under which conditions the latter Proposition \ref{prop:reverse} can be generalized for the mappings with countable fibers.

\section{$\Delta$-spaces vs properties of spaces $C_{p}(X)$}\label{Linear}
Our main goal here is to study the following question: \emph{Which topological properties related to being a $\Delta$-space are preserved by
the relation of $l$-dominance}?

The class of all distinguished lcs does not preserve continuous linear images.
To see this it suffices to consider the identical mapping from the Banach space $C[0,1]$ onto $C_p[0,1]$.
Below we show that the class of Tychonoff spaces $\Delta$ is invariant under the relation of $l$-dominance, equivalently, the class of distinguished $C_p$-spaces
is invariant under the operation of taking continuous linear images.

\begin{theorem}\label{linear1}
Assume that $Y$ is $l$-dominated by $X$.
 If $X$ is a $\Delta$-space, then $Y$ also is a $\Delta$-space.
\end{theorem}
For the reader's benefit, we present two different proofs of Theorem \ref{linear1}: topological and analytical ones.
In order to present the first proof, we start with the following simple lemma. Surprisingly, we were unable to find its formulation in any monograph
cited in the references.
By this reason we include its complete proof which rely on several extreme properties of the Tychonoff product $\R^X$.

\begin{lemma}\label{help}
Let $X$ and $Y$ be two sets and let $E \subset \R^X$ and $F\subset \R^Y$ be dense vector subspaces of $\R^X$ and $\R^Y$, respectively.
Assume that $T: E \longrightarrow F$ is a continuous linear surjection between lcs $E$ and $F$.
 Then $T$ admits a continuous linear surjective (unique) extension $\widehat{T}:\R^X \longrightarrow \R^Y$.
\end{lemma}
\begin{proof} Let us list all well-known properties of $\R^X$ we are going to use.\newline
\underline{Property 1}. Every closed vector subspace $H$ of $\R^X$ is complemented in $\R^X$ and the quotient
$\R^X/H$ is linearly homeomorphic to the product $\R^Z$ for some set $Z$ \cite[Corollary 2.6.5, Theorem 2.6.4]{bonet}.\newline
\underline{Property 2}. The product topology on $\R^X$ is minimal, i.e. $\R^X$ does not admit a weaker Hausdorff locally convex topology \cite[Corollary 2.6.5(i)]{bonet}. \newline
\underline{Property 3}. $\R^Y$ fulfills the extension property, i.e. if $M$ is a vector subspace of a lcs $L$, then every continuous linear mapping $T: M \longrightarrow \R^Y$
admits a continuous linear extension $\widehat{T}: L \longrightarrow \R^Y$ \cite[Theorem 10.1.2 (a)]{Narici}.

By Property 3, there exists a continuous linear extension $\widehat{T}: \R^X \longrightarrow \R^Y$ of $T$ such that $F\subset \widehat{T}(\mathbb{R}^{X})$.
 We prove that $\widehat{T}$ is a surjective mapping.
Denote by $\varphi:\mathbb{R}^{X}/\ker(T)\longrightarrow\mathbb{R}^{Y}$ the injective mapping associated with  the quotient mapping $Q: \mathbb{R}^{X}\longrightarrow\mathbb{R}^{X}/\ker(\widehat{T})$,
 where $\ker(\widehat{T})$ is the kernel of $\widehat{T}$ and $\varphi\circ Q = \widehat{T}$.
By Property 1, the space $\mathbb{R}^{X}/\ker(T)$ is linearly homeomorphic to the product $\mathbb{R}^{Z}$ for some set $Z$.
So we may assume that $\varphi$ is  a continuous linear bijection from $\mathbb{R}^{Z}$ onto a dense subspace $\widehat{T}(\mathbb{R}^{X})$ of $\mathbb{R}^{Y}$.
  This implies that on $\widehat{T}(\mathbb{R}^{X})$ there exists a stronger locally convex topology $\xi$ such that $(\widehat{T}(\mathbb{R}^{X}),\xi)$
	is linearly homeomorphic with $\mathbb{R}^{Z}$.
	However, by Property 2, $\mathbb{R}^{Z}$ does not admit  a weaker Hausdorff locally convex topology, hence $\widehat{T}(\mathbb{R}^{X})$
	is isomorphic to the complete lcs $\mathbb{R}^{Z}$. Finally, $\widehat{T}(\mathbb{R}^{X})$ is closed in $\mathbb{R}^{Y}$ and then $\widehat{T}$ is a surjection.
 \end{proof}
\begin{proof}[First Proof]
Let $T:C_{p}(X)\longrightarrow C_{p}(Y)$ be a continuous linear surjection. Denote by
 $\widehat{T}:\mathbb{R}^{X}\longrightarrow \mathbb{R}^{Y}$ the extension of $T$ which is supplied by Lemma \ref{help}.
By Theorem \ref{Theor:description}, $C_{p}(X)$ is distinguished and we can apply item (2) of Theorem \ref{three}.
Take arbitrary $f\in\mathbb{R}^{Y}$. There exists $g\in\mathbb{R}^{X}$ with $\widehat{T}(g)=f$. Then there exists a bounded set $B\subset C_{p}(X)$
such that  $g\in \cl_{\R^X}(B)$. We define $A = T(B)$. It is easy to see that $A$ is bounded and
$f\in \cl_{\R^Y}(A)$ which means that $C_{p}(Y)$ is distinguished, equivalently, $Y$ is a $\Delta$-space, by Theorem  \ref{Theor:description}.
\end{proof}

\begin{proof}[Second Proof]
If $T:C_{p}(X)\longrightarrow C_{p}(Y)$ is a continuous linear surjection, then by \cite[Proposition 23.30, Lemma 23.31]{meise},
the adjoint mapping  $T^*: (L_p(Y),\beta_{Y})\longrightarrow (L_p(X),\beta_{X})$ is continuous and injective,
 where $\beta_{X}$ and $\beta_{Y}$ are the strong topologies on the duals $L_{p}(X)$ and $L_{p}(Y)$, respectively.
Denote by $Z=T^*(L_p(Y))$. Endow $Z$ with the induced topology $\beta_X\hskip-2.5pt\restriction_Z$. Since
$T^*: (L_p(Y),\beta_{Y})\rightarrow (Z,\beta_{X}|Z)$ is a continuous linear bijection, the sets $T^*(U)$, where $U$ run over all absolutely convex neighbourhoods of zero in
$(L_p(Y),\beta_{Y})$, form a base of absolutely convex neighbourhoods of zero  for a locally  convex topology $\xi$ on $X$ such that  $\beta_{X}\hskip-2.5pt\restriction_Z\leq \xi$ and
 $T^*: (L_p(Y),\beta_{Y})\longrightarrow (Z,\xi)$ is a  linear homeomorphism.
Since $C_{p}(X)$ is distinguished by Theorem \ref{Theor:description}, the topology  $\beta_X$  is the  finest locally convex topology, by item (3) of Theorem \ref{three}.
 The property of having the finest locally convex topology is inherited by vector subspaces, so the induced topology  $\beta_{X}\hskip-2.5pt\restriction_Z$  is the finest locally convex one.
 Then  $\beta_{X}\hskip-2.5pt\restriction_Z=\xi$ is the finest locally convex topology, so $\beta_Y$ is of the same type on $L_{p}(Y)$.
Hence $C_{p}(Y)$ is distinguished, by Theorem \ref{three}, equivalently, $Y$ is a $\Delta$-space, by Theorem  \ref{Theor:description}.
\end{proof}

If $C_p(X)$ is homeomorphic to a retract of $\R^\kappa$ for some cardinal $\kappa$, then $X$ is discrete \cite[Problem 500]{Tkachuk_book2}.
Nevertheless, there exists a continuous mapping from $\R^\omega$ onto $C_p[0,1]$ \cite[Problem 486]{Tkachuk_book2}.
Several open problems have been posed in the following direction: Suppose that a dense subspace of $C_p(X)$ is a "nice" (not necessarily linear)
continuous image of $\R^\kappa$,
for some cardinal $\kappa$; must $X$ be discrete? \cite[Section 4.2]{Tkachuk_book1}.
Lemma \ref{help} implies immediately

\begin{corollary}\label{discrete}
Let a dense subspace of $C_p(X)$ be a continuous linear image of $\R^\kappa$,
for some cardinal $\kappa$. Then $X$ is discrete.
\end{corollary}

For simplicity, a topological space $X$ is called a \emph{$Q$-space} if each subset of $X$ is $F_{\sigma}$, or, equivalently,
each subset of $X$ is $G_{\delta}$ in $X$.

\begin{theorem}\label{linear2}
Let $X$ and $Y$ be normal spaces and assume that $Y$ is $l$-dominated by $X$.
 If $X$ is a $Q$-space, then $Y$ also is a $Q$-space.
\end{theorem}
\begin{proof} Normal $X$ is a $Q$-space if and only $X$ is \emph{strongly splittable}, i.e. for every $f\in\R^X$ there exists a sequence
$S=\{f_n: n\in\omega\} \subset C_p(X)$ such that $f_n \rightarrow f$ in $\R^X$, by \cite[Problems 445, 447]{Tkachuk_book3}.
Let $T: C_{p}(X) \longrightarrow C_{p}(Y)$ be a continuous linear surjection.
Denote by $\widehat{T}:\mathbb{R}^{X}\longrightarrow \mathbb{R}^{Y}$ the extension of $T$ which is supplied by Lemma \ref{help}.
Take arbitrary $f\in\mathbb{R}^{Y}$. There exists $g\in\mathbb{R}^{X}$ with $\widehat{T}(g)=f$. Then there exists a sequence $B\subset C_{p}(X)$
converging to $g$ in $\R^X$.
We define $A = T(B)$. It is easy to see that $A \subset C_p(Y)$ converges to $f$ in $\R^Y$.
\end{proof}

\begin{corollary}\label{Q-set}
Let $X$ and $Y$ be metrizable spaces (in particular, subsets of $\R$) and assume
that $Y$ is $l$-dominated by $X$.
If $X$ is a $Q$-set, then $Y$ also is a $Q$-set.
\end{corollary}

\begin{remark} Note that Theorems \ref{linear1} and \ref{linear2}, and Corollary \ref{Q-set} are valid
under a weaker assumption that a dense subspace of $C_p(Y)$ is a continuous linear image of $C_p(X)$.
\end{remark}

A space $X = \bigcup_{n\in\omega} X_n$ is called \emph{$\sigma$-scattered} (\emph{$\sigma$-discrete}) if every
$X_n$ is scattered (discrete, respectively).

\begin{proposition}\label{linear3}
Assume that $Y$ is $l$-dominated by $X$.
If $X$ is $\sigma$-scattered ($\sigma$-discrete), then $Y$ also is $\sigma$-scattered ($\sigma$-discrete, respectively).
\end{proposition}
\begin{proof}
Our argument is a modification of the proof of \cite[Theorem 3.4]{LLP} and is based
on an analysis of the dual spaces (see also \cite[Proposition 2.1]{Kawamura}).
Recall that for a Tychonoff space $X$, $L_{p}(X)$ denotes the dual space, that is, the space of all continuous
linear functionals on $C_{p}(X)$ endowed with the pointwise convergence topology. For each natural $n\in \N$
consider the subspace $A_n(X)$ of $L_{p}(X)$ formed by all words of the reduced length precisely $n$.
It is known that $A_n(X)$ is homeomorphic to a subspace of the Tychonoff product $(\R^*)^n\times X^n$, where $\R^*= \R\setminus\{0\}$.
Let $T: C_{p}(X) \longrightarrow C_{p}(Y)$ be a continuous linear surjection.
The adjoint mapping $T^*$ embeds $L_p(Y)$ into $L_p(X)$. Therefore, $Y$ can be represented as a countable union of
subspaces $Y_i, i \in \N$, such that each $Y_i$ is homeomorphic to a subspace of $(\R^*)^n\times X^n$ for some $n=n(i)$.

Consider the projection $p_i$ of each of the above pieces $Y_i \subset (\R^*)^n\times X^n$ to the second factor $X^n$.
The surjectivity of the linear mapping $T$ implies that $p_i: Y_i \longrightarrow X^n$ is a finite-to-one mapping.
Evidently, $X^n$ is scattered /discrete provided $X$ is. Since $p_i$ is continuous, for every isolated point $z \in X^n$ its finite fiber
$p_i^{-1}(z)$ consists of points isolated in $Y_i$ and the claim follows.
\end{proof}

The following question remained open.
\begin{problem} \label{prob5}
Let $Y$ be $l$-dominated by a scattered space $X$. Must $Y$ be scattered?
\end{problem}

Below we answer Problem \ref{prob5} positively in several interesting particular cases with the help of the properties of $\Delta$-spaces.

\begin{proposition}\label{prop:completely}
Let $X$ and $Y$ be metrizable spaces and assume that $Y$ is $l$-dominated by $X$.
If $X$ is scattered, then $Y$ also is scattered.
\end{proposition}
\begin{proof} If $X$ is metrizable and scattered, then $X$ is a $\Delta$-space by \cite[Proposition 4.1]{KL}.
Hence by Theorem \ref{linear1} the space $Y$  is a $\Delta$-space.
From another hand, every metrizable and scattered space is completely metrizable, by \cite[Corollary 2.2]{Tkachuk-1}.
A metrizable space $Y$ is $l$-dominated by a completely mertizable space $X$, therefore $Y$ is completely metrizable by the main result of \cite{Pelant}.
Finally, $Y$ is a \v{C}ech-complete $\Delta$-space, and $Y$ is scattered applying \cite[Theorem 3.4]{KL}.
\end{proof}

If $X$ and $Y$ both are compact spaces and there is a continuous mapping from $C_p(X)$ onto $C_p(Y)$, then $Y$ is Eberlein whenever $X$ is (see \cite[Theorem IV.1.7]{Arch5}),
and $Y$ is Corson whenever $X$ is (see \cite[Theorem IV.3.1]{Arch5}).
 Our next statement is a combination of a few known results,
while we apply Theorem \ref{linear1} in order to obtain the scatteredness of a target space.

\begin{proposition}\label{prop:Eber}
Assume that $Y$ is $l$-dominated by $X$.
\begin{enumerate}
\item[{\rm (1)}] If $X$ is an Eberlein compact, then $Y$ also is an Eberlein compact.
\item[{\rm (2)}] If $X$ is a scattered Eberlein compact, then $Y$ also is a scattered Eberlein compact.
\end{enumerate}
\end{proposition}
\begin{proof} (1)
The space $C_p(X)$ contains a dense $\sigma$-compact subspace, by \cite[Theorem IV.1.7]{Arch5}, hence $C_p(Y)$ satisfies the same property and
consequently, $C_{p}(Y)$ contains a compact subset $K$ which separates points of $Y$.
On the other hand, $Y$ is pseudocompact by the result of V. Uspenskii (see \cite{Arch4}).
By means of evaluation mapping we define a continuous injective mapping $\varphi: Y  \longrightarrow C_p(K)$.
Denote by $B=\varphi(Y)$. Then $B$ is a pseudocompact subspace of $C_p(K)$. Applying \cite[Theorem IV.5.5]{Arch5} we get that $B$ is an Eberlein compact.
 We showed that the pseudocompact space $Y$ is mapped by a continuous injective mapping $\varphi$ onto the Eberlein compact $B$.
However, the mapping $\varphi$ must be a homeomorphism, by \cite[Theorem IV.5.11]{Arch5} and the result follows.

(2) Every scattered Eberlein compact is a $\Delta$-space, by \cite[Theorem 3.7]{KL}.
Hence by above Theorem \ref{linear1}, $Y$ is a $\Delta$-space. We conclude that $Y$ is scattered, by \cite[Theorem 3.4]{KL}.
\end{proof}

\begin{remark}\label{Rez}
Proposition \ref{prop:Eber}(1) is not valid for Corson compacts.
E. Reznichenko showed that there exists a compact space $X$ with
the following properties (see \cite{Arch5}, \cite[Problem 222]{Tkachuk_book3}):
\begin{enumerate}
\item[{\rm (i)}] $C_p(X)$ is a $K$-analytic space, i.e. $X$ is a Talagrand (hence, Corson) compact;
\item[{\rm (ii)}] there is $x \in X$  such that $Y= X \setminus \{x\}$ is pseudocompact and $X$ is the Stone-\v{C}ech compactification of $Y$.
\end{enumerate}
Evidently, the restriction mapping projects continuously $C_p(X)$ onto $C_p(Y)$.
\end{remark}

\begin{remark}\label{Mar}
The assumption of linearity of continuous surjection between function spaces, even for compact spaces $X$ and $Y$,
 cannot be dropped in the main Theorem \ref{linear1} and its corollaries above.
Let $Y$ be any non-scattered metrizable compact, for instance $Y = [0,1]$. Denote by $S$ the convergent sequence.
Using the argument presented in \cite[Proposition 5.4]{Krupski}, one can construct a continuous surjective mapping from $C_{p}(S)$ onto $C_{p}(Y)$ (see also \cite[Remark 3.4]{Kawamura}).
\end{remark}

\begin{problem}\label{uniform}
 Assume that $Y$ is $u$-dominated by $X$.
 Is it true that $Y$ is a $\Delta$-space provided $X$ is a $\Delta$-space?
\end{problem}

Theorem \ref{linear1} may suggest also the following questions.
Below $C_{k}(X)$ stands for the space of all real-valued continuous functions on a Tychonoff space $X$
 endowed with the compact-open topology.
\begin{problem}\label{prob6}
 Assume that $X$ and $Y$ are Tychonoff spaces and there exists a continuous linear surjection from $C_{k}(X)$ onto $C_{k}(Y)$.
 Is it true that $Y$ is a $\Delta$-space provided $X$ is a $\Delta$-space?
\end{problem}

In case when the answer to Problem \ref{prob6} is negative one can pose the following

\begin{problem}\label{prob7}
Find scattered compact spaces $X$ and $Y$ such that $X\in\Delta$ but $Y\notin\Delta$
and there exists a continuous linear surjection from the Banach space $C(X)$ onto the Banach space $C(Y)$.
\end{problem}
Surely, for such $X$ and $Y$ a continuous linear surjection from $C_{p}(X)$ onto $C_{p}(Y)$ does not exist, by Theorem \ref{linear1}.
Notice also that $X$ in Problem \ref{prob7} cannot be an Eberlein compact, since otherwise $Y$ would be a scattered Eberlein compact, hence $Y$ would be in the class $\Delta$.

\begin{proposition}\label{Cech-complete}
Let $X$ be a \v{C}ech-complete Lindel\"of space. Then
the following assertions are equivalent.
\begin{enumerate}
\item[{\rm (1)}] $X$ is scattered.
\item[{\rm (2)}] $X$ is $\sigma$-scattered.
\item[{\rm (3)}] $C_{p}(X)$ is a Fr\'echet-Urysohn space.
\end{enumerate}
\end{proposition}
\begin{proof} The implication (2) $\longrightarrow$ (1) follows from the well-known fact that
every closed subspace of $X$ satisfies Baire category theorem.
The equivalence (1) $\longleftrightarrow$ (3) has been proved already in \cite[Corollary 2.12]{ga-ka}.
\end{proof}

\begin{corollary}
Let $X$ be a \v{C}ech-complete Lindel\"of space. If $X \in \Delta$, then $C_{p}(X)$ is a Fr\'echet-Urysohn space.
\end{corollary}

Let us call a lcs $E$ \emph{hereditarily distinguished} if every closed linear subspace of $E$ is distinguished.
It is known that even a Fr\'echet distinguished lcs can contain a closed non-distinguished subspace.
The only hereditarily distinguished $C_p$-spaces we are aware of are the products of reals $\R^{\kappa}$.
Note that if $\varphi$ is a continuous mapping from a compact $\Delta$-space $X$ onto $Y$, then
the adjoint mapping $\varphi^*$ identifies $C_p(Y)$ with a closed linear subspace of $C_p(X)$,
and this closed copy of $C_p(Y)$ is again distinguished. Our last problem is inspired by this observation.

\begin{problem}\label{prob8}
Does there exist an infinite compact space $X$ such that $C_p(X)$ is hereditarily distinguished?
More specifically, let $X$ be the one-point compactification of an infinite discrete space. Is $C_p(X)$ hereditarily distinguished?
\end{problem}

\section{Compact $\Delta$-spaces and PFA}\label{PFA}
A topological space $X$ has countable tightness if for each $A \subset X$ and for each $x \in \cl(A)$, there is a countable $B \subset A$ such that $x \in \cl(B)$.
A topological space $X$ is a sequential space if $A \subset X$ is a sequentially closed set implies that $A$ is closed.
 The set $A$ is a sequentially closed set if a countable sequence $(x_n) \in A$ converges to $x \in X$ implies that $x \in A$.
Every sequential space is countably tight.
A topological space $X$ is called \emph{$\omega$-bounded} if the closure of every countable subset of $X$ is compact.
$X$ is called \emph{pseudocompact} if every continuous function defined on $X$ is bounded.
Evidently, every $\omega$-bounded space is countably compact, and every countably compact space is pseudocompact.
The space of countable ordinals $[0,\omega_{1})$ is an example of an $\omega$-bounded space which is not compact.
 A continuous mapping $f: X \longrightarrow Y$ is called perfect if
it is closed and $f^{-1}(y)$ is compact for each $y\in Y$.

\begin{theorem}\label{theor:omega}
Every $\omega$-bounded $\Delta$-space is compact.
\end{theorem}
\begin{proof} Assume that $X$ is a counter-example to the claim. Then, by a result of D. Burke and G. Gruenhage \cite[Lemma 1]{Gruenhage},
$X$ contains a subset $Z$ which is a perfect preimage of the ordinal space $[0,\omega_{1})$. We conclude that a $\Delta$-space $Z$ can be mapped by a continuous closed mapping onto
$[0,\omega_{1})$. By our Proposition \ref{prop:closedmap} this would mean that $[0,\omega_{1}) \in \Delta$,  however, the opposite is true \cite{KL}.
Obtaining contradiction finishes the proof.
\end{proof}

It has been shown in \cite{LS} that every compact $\Delta$-space has countable tightness.
Essentially the same argument as in Theorem \ref{theor:omega} provides a very short proof of this assertion.

\begin{theorem}\label{tightness}\cite{LS}
Every compact $\Delta$-space has countable tightness.
\end{theorem}
\begin{proof} A compact space has countable tightness if and only if it does not contain a perfect preimage of $[0,\omega_{1})$ (see \cite{Balogh}).
We argue again that every $\Delta$-space satisfies this property.
\end{proof}

A very natural question arises whether Theorems \ref{theor:omega} and \ref{tightness} can be generalized for countably compact spaces.
A positive answer follows from the Proper Forcing Axiom (PFA), due to the celebrated results of Z. Balogh \cite{Balogh} (see also \cite{BDFN}).

\begin{theorem}\label{theor:PFA}
{\rm (PFA)}
\begin{enumerate}
\item[{\rm (1)}] Every countably compact $\Delta$-space is compact.
\item[{\rm (2)}] Every countably compact $\Delta$-space has countable tightness.
\item[{\rm (3)}] Every countably compact $\Delta$-space (hence, every compact $\Delta$-space) is sequential.
\end{enumerate}
\end{theorem}

\begin{problem}\label{prob9}
Is it possible to obtain the results of Theorem \ref{theor:PFA} in ZFC alone?
\end{problem}

Note that all known examples of compact $\Delta$-spaces are $\sigma$-discrete. However, we don't know
if it is always the case.

\begin{problem}\label{prob3}
Let $X$ be a compact $\Delta$-space. Is $X$ a $\sigma$-discrete space?
\end{problem}

A closely related question to the last problem is the following one: When a $\Delta$-space is scattered?
 As we have been mentioned earlier every \v Cech-complete $\Delta$-space
 (in particular, every compact $\Delta$-space) is scattered \cite{KL}.

\begin{example}\label{ex:Baire} There exists a Baire countable space which is not scattered.
Fix in the real line $\R$ a countable dense subset $B$ consisting of irrationals. Let $X$ be the union of the rationals $\Q$ with $B$.
Equip $X$ with the topology inherited from the Michael line $M$. Then $X$ is a countable space containing a copy of $\Q$, therefore $X$ is a non-scattered $\Delta$-space.
$X$ is Baire since it contains a dense discrete subspace.
\end{example}

Despite of Problem \ref{prob9} the following result does not require extra set-theoretic assumptions.

\begin{theorem}\label{th:count_compact} Every countably compact $\Delta$-space is scattered.
\end{theorem}
\begin{proof} On the contrary, assume that a countably compact space $X$ is not scattered.
Every countably compact space $X$ is pseudocompact, therefore there exists a closed subset $K \subset X$ and a continuous surjective mapping $\varphi$
from $K$ onto the closed interval $[0,1]$, by \cite[Proposition 5.5]{LT}. Every closed subset $F$ of $K$ is a countably compact space, its continuous image
$\varphi(F)$ is a countably compact subset of $[0,1]$, therefore $\varphi(F)$ is compact. We conclude that $\varphi\,\res\,{K}$ is a closed continuous mapping from $K$ onto $[0,1]$.
This evidently contradicts Theorem \ref{th:F-sigma_map}, since $[0,1]\notin\Delta$.
\end{proof}

The proof above fails if we assume only that $X$ is a pseudocompact (and non-normal) space, in view of Proposition \ref{prop:locomp}.

\begin{problem}\label{prob4} Prove that every pseudocompact $\Delta$-space is scattered, thereby prove that a Tychonoff space $Y$ is scattered provided it is
$l$-dominated by a compact $\Delta$-space $X$.
\end{problem}
\textbf{Acknowledgments.} The authors thank Witold  Marciszewski for the useful information about Corson and Eberlein compact spaces.

The authors acknowledge and thank Vladimir Tkachuk for the most stimulating letters \cite{Tkachuk}.

redpajama_set_name: RedPajamaArXiv


text:
Phoenix Bar rises as new, suburban Chicago gay bar and nightclub
Wed. September 18, 2013 8:53 AM
#Business #Local
justin romme, phoenixbar general manager
photo credit // phoenixbar/progressbar
Phoenix Bar replaces Hunters Chicago
Chicago, IL - Sometime in early to mid October, a familiar home for thousands of Chicago area gays will get a new look and a new name, when the real estate and assets of the legendary Hunter's Nightclub in suburban Elk Grove Village will transfer to new ownership and management.
The new nightclub, Phoenix Bar & Nightclub, has some big shoes to fill. For more than 30 years, Hunter's has been the premier gay nightclub in the Chicago suburbs. Its central location at 1932 E. Higgins Road is only minutes from all of Chicagoland via I-90, and is only minutes from O'Hare.
"I count myself among so many for whom Hunter's was their first gay bar experience," said the new PhoenixBar General Manager Justin Romme. "It's a huge space, with a great vibe and an incredible staff."
Phoenix Bar is owned and managed by the same team that opened the popular Progress Bar this past June, at the corner of Halsted and Roscoe in the heart of Chicago's Boystown neighborhood.
Romme worked with Progress Bar's creator and managing partner David Sikora for the past few months, after several years working for Hilton Corporation. Both bars have the same vision statement—"where everyone feels important."
"Management, employees, customers and community—we are singularly driven to provide an environment of inclusion that's welcoming and supportive to everyone," Sikora said. "The response at ProgressBar has been overwhelmingly positive. When we heard that the Hunter's space was for sale, we acted immediately—the best gay bar in the suburbs, hands down."
The new owners recognize there is a huge legacy to be preserved in the 6,500 square foot space. With a capacity of nearly 600 (more with the popular outside patio from May through October), a great sound and light system, a sprawling dance area and 150 free parking spaces. "It's no wonder that the space has been a community destination for 30 years," commented Romme.
"There are so many great things about the current space....and we are keeping the vast majority of the current Hunter's staff, because they are absolutely top notch and have a loyal following." Romme said. "But the name 'Phoenix' implies a new beginning. We are revamping the interior, updating the look and feel of the space. Lots of brand new giant flatscreens, new interior design elements, new colors."
But Romme is particularly excited about Phoenix Bar's new, custom-designed dance floor, slated to be completed a few weeks after the bar's opening. "It's essentially a giant flat-screen television, constructed of multiple seamless video panels. There's nothing like it in the Chicago area. We hope to bring the bar's dance club tradition to a whole new level."
With an open date in early to mid-October, after a few weeks to complete an intense interior makeover of the space, Phoenix Bar will open "with a whole new look and whole new attitude" according to Romme. "This space is legendary. But once we've completed our extensive interior makeover, we will strive to be the new premier gay bar and nightclub in Chicago's suburbs. Exciting weekly promotions and nights, coupled with a great new look and an incredible staff... truly, it will be Phoenix rising."
Golden-Con response overwhelming, April 'Golden Girls' fan convention moved to Chicago's Navy...
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Trending: Golden-Con grows, moves to Navy Pier; HRC, senators call on FDA...
HRC, Senators call on FDA to lift gay, bisexual blood ban

redpajama_set_name: RedPajamaCommonCrawl


text:
angular
	.module(`${window.APP_NAME}.pages`)
	.directive('videoPage', videoPage);

const TEMPLATE = `

			{{ctrl.video.teams.teams[0].name}}
			vs.
			{{ctrl.video.teams.teams[1].name}}

			Mysterious match

				Video not found

				Sorry, we cannot find video with that id

		<video object="ctrl.video" show-hotkey-info="true" ng-if="!ctrl.error"></video>

`;

/**
 * @emits Constants.Event.TournamentSelectedEvent(tournament:Object)
 * @emits Constants.Event.CastersSelectedEvent(casters:Array<Object>)
 */
function videoPage () {

	return {
		restrict: 'E',
		scope: true,
		template: TEMPLATE,
		controller: controller,
		controllerAs: 'ctrl'
	};

	function controller ($http, $rootScope, $state, Page, Constants) {
		var self = this;

		self.video = {};

		$http.get(`${Constants.Api.VIDEO}/${$state.params.id}`)
			.then(response => {
				self.video = response.data;
				Page.loaded();
				setTitle(self.video);

				$rootScope.$broadcast(Constants.Event.TournamentSelectedEvent, self.video.tournament);
				if (self.video.casters.casters) {
					$rootScope.$broadcast(Constants.Event.CastersSelectedEvent, self.video.casters.casters);
				}
			})
			.catch(response => {
				self.error = response.data;
				Page.loaded();
				Page.setTitle('Not Found');
			});

		function setTitle (video) {
			if (!video.teams || !video.teams.teams || !video.teams.teams.length) {
				if (video.tournament.name) Page.setTitle(video.tournament.name);
				else Page.setTitle('Mysterious match');
			}
			else Page.setTitle(`${video.teams.teams[0].name} vs ${video.teams.teams[1].name}`);
		}
	}

}

redpajama_set_name: RedPajamaGithub


Just like drinkers celebrating the impending end of dry January, these adorable fawns have been filmed guzzling milk like there's no tomorrow.
As a volunteer at a wildlife sanctuary arrives with bottles of milk, the eight young deer practically pounce on the teets and begin drinking furiously.
One fawn, who drinks from a bottle held the man, has to be helped to manoeuvre his legs so he can reach the milk.
Founded in 2011, Nora's Ark responds to the rehabilitation needs of those animals that are sick, injured, or orphaned and release healthy wildlife back into their natural habitat.
Later in life, these fawns will feed primarily on leaves.
Deers tend to eat eat easily digestible shoots, young leaves, fresh grasses, soft twigs, fruit, fungi, and lichens.
Responses to "Herd of fawns rush to drink from bottles of milk at Wildlife sanctuary"


text:
Q: Geting SQL Server instances names from registry I have tried to get my SQLServer instances names from the registry by using this code:
type
  TRegistryHelper = class helper for TRegistry
  public
    function ReadMultiSz(const name: string; var Strings: TStrings): boolean;
  end;

function TRegistryHelper.ReadMultiSz(const name: string;
  var Strings: TStrings): boolean;
var
  iSizeInByte: integer;
  Buffer: array of WChar;
  iWCharsInBuffer: integer;
  z: integer;
  sString: string;
begin
  iSizeInByte := GetDataSize(name);
  if iSizeInByte > 0 then begin
    SetLength(Buffer, Floor(iSizeInByte / sizeof(WChar)));
    iWCharsInBuffer := Floor(ReadBinaryData(name, Buffer[0], iSizeInByte) / sizeof(WChar));
    sString := '';
    for z := 0 to iWCharsInBuffer do begin
      if Buffer[z] <> #0 then begin
        sString := sString + Buffer[z];
      end else begin
        if sString <> '' then begin
          Strings.Append(sString);
          sString := '';
        end;
      end;
    end;
    result := true;
  end else begin
    result := false;
  end;
end;

procedure TForm1.Button1Click(Sender: TObject);
const
  cKey = '\SOFTWARE\Microsoft\Microsoft SQL Server';
var
  Registry: TRegistry;
  MyList: TStrings;
begin
  Registry := TRegistry.Create;
  Registry.RootKey := HKEY_LOCAL_MACHINE;
  if Registry.OpenKeyReadOnly(cKey) then
  try
    MyList := TStringList.Create();
    Registry.ReadMultiSz('InstalledInstances', MyList);
    ListBox1.Items.Assign(MyList);
  finally
    MyList.Free;
  end;
  Registry.Free;
end;

But I have noted the iSizeInByte = -1 every time, and I failed to get those names by this way.
Also, I have noted that when trying to get those instances from my TADOQuery connection string builder, the component also failed to get those names.
Is there any way to get them?

redpajama_set_name: RedPajamaStackExchange


text:
Home / Politics / Agenda 50/50 "Ky3 Mu P3" In Ashanti Region – Former Kumasi Mayor Assures
Mr. Kojo Bonsu, the former Kumasi Metropolitan Assemble Mayor
Agenda 50/50 "Ky3 Mu P3" In Ashanti Region – Former Kumasi Mayor Assures
admin June 28, 2022 Politics Leave a comment 189 Views
Mr. Kojo Bonsu, the former Kumasi Metropolitan Assemble Mayor, has this morning reignited the fire and zeal in members and sympathizers of the National Democratic Congress (NDC) as he assured them of splitting the political fortunes within the Ashanti Region among the NDC and NPP.
He gave this refreshing assurance on Angel Tv/FM's flagship morning show "Anopa B)fo)" with Nana Yaa Brefo and her co-hosts.
The former Kumasi Mayor said he is currently the only person who has the capacity to break the long held perception that Ashanti Region is solely the comfort zone of the NPP.
"A lot of people still hold the view that Ashanti Region is only for the NPP party but I say it is false. NDC can break that record and I, Kojo Bonsu is the only person who can divide the votes in the Ashanti Region for the NDC to get the maximum needed votes to win the 2024 elections. That is why I am working on what I call the agenda 50/50, "Ky3 Mu p3″ where we the NDC take 50% of the votes in Ashanti Region and the other 50% for the NPP and I am the only person who can lead the NDC to acheive this feat", he assured.
See also 2024 Polls: NDC's New Leaders Are Very Experienced Politicians; We Have A Long Way To Go – NPP Kwasi Korang Writes
When asked if he will still pursue his agenda 50/50 should the NDC vote for a different presidential candidate, the former National Sports Authority (NSA) boss said his agenda still remains regardless of the outcome of the elections. Mr Bonsu is convinced the delegates would endorse him however in the unlikely event that the endorsement goes to another person, he will still stand firmly behind the chosen candidate, mobilize his base and rally the support of all to accomplish his mission of dividing the Ashanti region votes for the sake of victory in 2024.
"I am very convinced the NDC will choose me as the next presidential candidate because the reality of the need for change is a fact known and accepted by the majority of Ghanaians today and my beloved party, the NDC should have learnt many painful but important lessons from the eight years we have endured in opposition to make the right decisions that can change the party's destiny. Even in the unlikely event that it goes to another candidate, I will support that party member and make sure that we divide the Ashanti region votes by half to win the 2024 general elections". He added.
See also Illegal Mining: President Akufo-Addo Should Have Resigned By Now – Mahama
Responding to a question on whether or not he would accept a running mate slot from anyone else who wins the NDC race, he said "that time has not come yet. In planning to take power from the NPP or any party for that matter, you must plan well and excite flawlessly. At this juncture we are working towards the priority objective which is to lead the party however if by the end of this year we find ourselves in such a situation as you describe then the bottom line would be my willingness to serve the party in any capacity required and I have always made that clear not only in words but in deed" he explained.
See also George Opare Addo, Hanna Bissiw Retained As NDC's Youth And Women's Organisers [See Full Results]
Mr. Kojo Bonsu, is one of the few individuals in the NDC party who has hinted strongly of his intention to contest the party's presidential candidate position when the official nomination process begins. In line with his innovative and passionate spirit he has since last year led the call for the reorganization of the party, popularly dubbed "Reboot for Victory" to energize the party's base to secure victory in 2024 for All Ghanaians.
Ashanti Region Mr. Kojo Bonsu NDC 2022-06-28
Tags Ashanti Region Mr. Kojo Bonsu NDC
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text:
BMW to use Neue Klasse platform for 3-Series segment full EVs
The platform will play 'a crucial part' in the ramp up of EV sales for the company, CEO Oliver Zipse said.
The next-generation 3 Series will have a full-electric version. Shown is the current 320e sedan.
BMW will initially focus its Neue Klasse (New Class) architecture on full-electric models for the midsize premium segment, CEO Oliver Zipse said.
The platform, which will go into production in 2025 at the automaker's new factory in Hungary, will only offer a battery-electric drivetrain, Zipse said during the automaker's first-quarter earnings call on Thursday.
"When it hits the market, it will be concentrated on the 3-Series segment and at that point in time the market will have developed into a size where it is reasonable to have only one drivetrain in that architecture," Zipse said.
Zipse's comments highlight BMW's change in thinking at the automaker about the architecture, which originally had been planned to be fully electric but also underpin cars with gasoline and diesel internal combustion engines, including plug-in hybrids.
BMW said at the IAA Munich auto show in September that the Neue Klasse would be electric-only.
On Thursday, Zipse said the architecture will be "mainly focused on the middle car (midsize) segment. It does not cover from the lower segments all the way to the luxury segments," he said.
However, after its initial launch on midsize cars, BMW plans to expand the architecture's use for all-electric cars in other segments.
"The Neue Klasse starts with models from the middle car segment, but of course there is more to come," a BMW spokesman told Automotive News Europe. "The Neue Klasse is our model range for the future – from the high-volume segment to exclusive high-performance models," the spokesman said.
The strategy shift to use the platform only on full-electric midsize models will allow BMW to keep combustion engines for longer on larger models.
BMW is launching a new generation of its 7-Series flagship sedan this year with plug-in hybrid, diesel and gasoline drivetrains, and a full-electric variant badged as the i7.
Zipse said on the earnings call that BMW aimed to hit cumulative sales of 2 million full-electric cars by 2025 on its way to a target of having half its sales being electric-only by 2030.
"We are doing everything we can to meet this goal earlier," he said.
The Neue Klasse platform will play "a crucial part" in the ramp up of EV sales for the company, Zipse said: "It represents a quantum leap in technology."
The architecture will have the automaker's next-generation electric drivetrain "with more output, new cell chemistry and new cell formats," Zipse said.
Zipse's clarification that the platform will be reserved only for battery-electric vehicles (BEV) was noted by analysts at Deutsche Bank.
"We view it as positive that BMW says that from 2025 onward the Neue Klasse will be BEV only, while a lot of market participants had so far understood it was BEV first," the bank wrote in a note to investors.
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Green? For the festive season? I know, groundbreaking! But, let's be honest, how excited are we all for rich reds, metallics and Christmas-tree green to come back into our wardrobe? These amazing colours are definitely on their way back in. But there are a few other trends, hues and textures that I want to chat to you about for the gazillion Christmas parties you've got lined up in your iCal.
Let's kick off with the classic green, but not as you know it. This season's top piece that everyone and their Insta profile has been rocking, the midi skirt, has had a wintery update with this amazing colour. This will look awesome with chunky knitwear and boots for daytime and a cute cami and heels for the evening.
Green doesn't get through the Christmas season without meeting it's bestie mate velvet. These velvet pants just scream Christmas to me and are the ultimate in festive sass.
Multi coloured sparkles are seriously big for this party season. Nasty Gal are absolutely killing it with this gorgeous long sleeved mini. Paired with the socks and sandals combination from above? You're onto a winning outfit. Plus, you can grab student discount on Nasty Gal with your UNiDAYS membership and you'll be laughin' all the way to the party at how much of a bargain your outfit was!
Topshop are getting in on the act with this incredible multi-toned green glittery dress and fancy tights combo from their #TOPSHOPXHALPERNcollection, giving off all the Studio 54 glamour.
Miss Selfridge are jumping on this rainbow sparkle trend with an awesome jumpsuit. This is perfect if you're not a fan of dresses for going out in and it also means that you don't have to bother fake-tanning your whole body. Wins all round. Plus, this is another dream that you can bag money off with your student status!
Another one of my favourite brands for affordable party wear, Dorothy Perkins are loving this trend with this easy to wear sequin tee. This is amazing if you love the classic 'jeans and a nice top' party look. Pop on your favourite black skinnies and your comfiest heels and you'll be the happiest queen at the party.
Finally on the multi-toned glitz is a gem from Missguided. This is beaut if you like the more neutral tones and rainbow is giving you the fear! Shop it and save with your UNiDAYS student discount, gal!
This Boohoo sporty sheer socks and gold platform sandals combo is just ridiculously cute and everyone needs to be wearing it, this instant! No more chilly toes trudging from party to party for you, no sir.
There we have my perfect trend-focused pieces for the festive season. Who'd have thought we'd be actually wanting to wear tights and socks for party season? Not me but I am all over it! Times have changed with these awesome hosiery updates. We've still got our safety blanket of green and sparkles but with a little unicorn-style edge. How could you not enjoy party time when you can dress up in these incredible pieces?


Born in Vancouver in 1952, Bruce Hugh Russell studied at the Vancouver School of Art (now Emily Carr University) and at Concordia University. An independent curator and art historian, he has written extensively on both contemporary art and art history, publishing in numerous journals including Canadian Art, Studies in Visual Communications, and Parachute, and has written for the Canadian Centre for Architecture, the Belkin Gallery at the University of British Columbia, The Ottawa Art Gallery, and the Montreal Museum of Fine Arts, among others.
Mr. Russell has enjoyed a long and varied relationship with the National Gallery of Canada, as a scholar, curator, and donor. In 1993-1994 he received the prestigious Resident Research Fellowship in Canadian Art History, at National Gallery of Canada, for research on the life and work of patron, collector and craftsman Douglas Duncan 1902-1968. He was a Contributor to the catalogue of the Gallery's exhibition, The Ingenious Machine of Nature: Four Centuries of Art and Anatomy in 1996, and more recently he was a co-curator of the exhibition Artists, Architects and Artisans: Canadian Art 1890–1918. In this exhibition he was able to draw on his long-term interest in the Arts & Crafts movement and in the decorative arts and architecture of the Anglican tradition.
His present projects include a major biography of Montreal cultural philanthropist F. Cleveland Morgan and a study of Saskatchewan-born abstract painter Agnes Martin's Canadian associations, (2018-2019 at the Esker Foundation, Calgary, and the MacKenzie Gallery, Regina.) Mr. Russell now lives in Regina, and has recently retired from teaching in the northern community of La Ronge where he taught art history to aboriginal under-graduate students.


text:
Most offices and departments responsible for oversight, operation, and maintenance of city-owned facilities and public services. are located in the Municipal Complex.
The City currently has 133 full-time employees, 35 part-time employees, and three seasonal employees.
Administrative/Finance and Building Maintenance: City Clerk-Treasurer, seven full-time accounting/clerical employees, two full-time custodial employees.
Executive: six Elected Officials (Mayor & Council members), one full-time clerical employee, one full-time human resources employee.
Code Enforcement: Superintendent, two full-time code Inspectors, one full-time clerical employee.
Fire Department: Fire Chief, 12 full-time firefighters, 16 part-time firefighters.
Municipal Court: Chief Court Clerk, four full-time court magistrates, contracted services of one judge, one prosecutor, and two public defenders.
Police Department: Police Chief, 38 full-time sworn officers, one full-time clerical employee, one full-time animal control employee (non-sworn), three seasonal school crossing guards.
Public Works and Animal Shelter: Superintendent, one full-time clerical employee, 25 full-time public works employees, two full-time animal shelter employees.
Vehicle Maintenance: Superintendent, one full-time clerical employee, five full-time mechanics.
Library: Director, nine full-time employees, one full-time maintenance employee, five part-time employees.
Parks & Recreation: Director, 11 full-time employees, 12 part-time employees.


text:
using glm::mat4;
using glm::vec3;
using glm::vec4;

namespace
{
const glm::vec3 CAMERA_EYE = { 0, 1, 2 };
const glm::vec4 SUNLIGHT_POSITION = {0, 0, 0, 1};
const glm::vec4 WHITE_RGBA = {1, 1, 1, 1};
const glm::vec4 FADED_WHITE_RGBA = {0.3f, 0.3f, 0.3f, 1.0f};

void SetupOpenGLState()
{
    // включаем механизмы трёхмерного мира.
    glEnable(GL_DEPTH_TEST);
    glEnable(GL_CULL_FACE);
    glFrontFace(GL_CCW);
    glCullFace(GL_BACK);
}

glm::mat4 MakeProjectionMatrix(const glm::ivec2 &size)
{
    // Матрица перспективного преобразования вычисляется функцией
    // glm::perspective, принимающей угол обзора, соотношение ширины
    // и высоты окна, расстояния до ближней и дальней плоскостей отсечения.
    const float fieldOfView = glm::radians(70.f);
    const float aspect = float(size.x) / float(size.y);
    const float zNear = 0.01f;
    const float zFar = 100.f;

    return glm::perspective(fieldOfView, aspect, zNear, zFar);
}
}

CWindowClient::CWindowClient(CWindow &window)
    : CAbstractWindowClient(window)
    , m_defaultVAO(CArrayObject::do_bind_tag())
    , m_keplerSystem(m_timeController)
    , m_rotationSystem(m_timeController)
    , m_mouseGrabber(window)
    , m_camera(CAMERA_EYE)
{
    const vec4 BLACK_RGBA = {0, 0, 0, 1};
    window.SetBackgroundColor(BLACK_RGBA);
    SetupOpenGLState();

    m_renderSystem.SetupLight0(SUNLIGHT_POSITION, WHITE_RGBA, FADED_WHITE_RGBA);

    CSceneLoader loader(m_world);
    loader.LoadScene("res/solar_system/solar_system_2012.json");
    loader.LoadSkybox("res/solar_system/skybox.plist");

    // Добавляем систему, отвечающую за изменение положения планет
    //  согласно их орбитам и прошедшему времени по законам Кеплера.
    m_world.addSystem(m_keplerSystem);

    // Добавляем систему, выполняющую вращение тел вокруг своих осей.
    m_world.addSystem(m_rotationSystem);

    // Добавляем систему, отвечающую за обновление систем частиц
    m_world.addSystem(m_particleUpdateSystem);

    // Добавляем систему, отвечающую за рендеринг планет.
    m_world.addSystem(m_renderSystem);

    // Добавляем систему, отвечающую за рендеринг систем частиц
    m_world.addSystem(m_particleRenderSystem);

    // После активации новых сущностей или деактивации,
    //  а при добавления новых систем следует
    //  вызывать refresh() у мира.
    m_world.refresh();
}

void CWindowClient::OnUpdate(float deltaSeconds)
{
    m_camera.Update(deltaSeconds);
    m_timeController.Update(deltaSeconds);
    m_keplerSystem.Update();
    m_rotationSystem.Update();
    m_particleUpdateSystem.Update(deltaSeconds);
}

void CWindowClient::OnDraw()
{
    const glm::ivec2 windowSize = GetWindow().GetWindowSize();

    const mat4 view = m_camera.GetViewMat4();
    const mat4 projection = MakeProjectionMatrix(windowSize);

    glViewport(0, 0, windowSize.x, windowSize.y);
    m_renderSystem.Render(view, projection);
    m_particleRenderSystem.Render(view, projection);
}

bool CWindowClient::OnKeyDown(const SDL_KeyboardEvent &event)
{
    return m_camera.OnKeyDown(event);
}

bool CWindowClient::OnKeyUp(const SDL_KeyboardEvent &event)
{
    return m_camera.OnKeyUp(event);
}

bool CWindowClient::OnMousePress(const SDL_MouseButtonEvent &event)
{
    return m_camera.OnMousePress(event);
}

bool CWindowClient::OnMouseMotion(const SDL_MouseMotionEvent &event)
{
    return m_mouseGrabber.OnMouseMotion(event)
            || m_camera.OnMouseMotion(event);
}

bool CWindowClient::OnMouseUp(const SDL_MouseButtonEvent &event)
{
    return m_camera.OnMouseUp(event);
}