# Tagged Questions

3answers
50 views

### Prove that $S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.

Prove that $S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$, but $\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}$ is not a compact set. (Can we use ...
1answer
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### Every countably compact, scattered $T_3$-space is sequentially compact

A space $X$ is called scattered provided every non-empty subspace $Y$ has an isolated point( with respect to the subspace topology on $Y$). How to prove that: Every countably compact, scattered ...
0answers
53 views

### Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
2answers
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### Is my proof correct? (minimal distance between compact sets)

I'm working out the following problem form Ahlfors' Complex Analysis text: "Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that ...
1answer
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### Understanding the topological space $\beta \omega$

It is from the Handbook of Set-theoretic topology: We consider $\beta \omega$ as the set of all ultrafilters on $\omega$ with the topology which is generated by taking as a base all sets of the ...
1answer
59 views

### Compactness of the set of terms in a convergent (sub)net and its limit

In a Hausdorff topological space, let net $(x_d)_{d \in D}$ converge to $x$. The set $$(\cup_{d \in D} \{x_d\}) \cup \{x\}$$ consisting of its terms and the limit need not be compact: in ...
0answers
129 views

### If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?

If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
0answers
84 views

### Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?

The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then \$|X| \ge ...
1answer
187 views

### One point compactification of $[0,1] \times [0,1)$

Let $X = [0,1] \times [0,1) \subset \mathbb{R}^2$. I've already proven that this space is locally compact and found its one-point compactification but now I am stuck on the following; Let \$Y = X \cup ...
1answer
140 views

### Any net in $A\subseteq X$ has a cluster point in $X$. Is $\overline{A}$ compact?

Let $(X,\mathcal T)$ be a topological space, $A\subseteq X$ and any net in $A$ have a cluster point in $X$. Is $\overline{A}$ compact?
2answers
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### How I can prove that: If $X$ is compact, then any map $f\colon X \to Y$ is proper?

A continuous map $f\colon X \to Y$ of locally compact spaces is called proper if for any compact $C\subset Y$ the preimage $f^{-1}(C)$ is compact. My question is: How I can prove that: If $X$ is ...
1answer
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### One question related with sequential compactness

While trying to prove that a set is sequentially compact, I was suggested to prove by contradiction -- this is how it went, at least part of it: Definition. We say that a set $A$ is sequentially ...
3answers
138 views

### Stone-Čech Compactification of the Natural numbers

I am trying to prove that if $U$ is contained in the Stone-Čech Compactification of the natural number ($\beta N$) that the closure of $U$ is open. I have a really hard time with even understanding ...
1answer
55 views

### A question on the compact subset of $[0,1]$

Let $S=\{K \subseteq [0,1]: K \text{ is compact and uncountable } \}$. How to see that $|S|=\mathfrak c$? Thanks for your help.
2answers
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### Inverse limit of compact metric spaces

Does an inverse limit of compact metric spaces need to be metrizable? When it is an inverse limit of a countable inverse system I know it is metrizable (even without compactness). But what if the ...
1answer
70 views

### Tychonoff Theorem and the axiom of choice

How to show that The Tychonoff Theorem and the axiom of choice are equivalent? Here I want to collect ways to prove it. Thanks for your help.
1answer
61 views

### If $X$ is complete and totally bounded, then $X$ is compact [closed]

Let $X$ be a metric space. Whar is your favorite way to show: If $X$ is complete and totally bounded, then $X$ is compact? Thanks for your help.
2answers
50 views

### A $T_2$ space is locally compact iff..

A $T_2$ space is locally compact iff it has a base $\beta$ s.t $\forall B\in \beta$ we have $\bar B$ is compact. My definition of locally compact is that $\forall x\in X$ has a compact neighborhood ...
2answers
63 views

### Question on compactification

I was studying for quals and had trouble with this question. Any help would be great, thanks. A two-point compactifcation of a Hausdorff space $X$ is a compact Hausdorff space $Y$ such that $X$ is a ...
0answers
110 views

### If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?

Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map. Using the embedding \$Y\hookrightarrow\beta ...
1answer
106 views

### Clopen subsets of a compact metric space

I am aked to show that in a compact metric space we can find at most countably many subsets which are both: open and close. I would be grateful for your help.
3answers
133 views

### $f:X\to X$ is one-one and continuous on a compact space. Is $f$ surjective?

Let $(X,\mathcal T)$ be a compact Hausdorff topological space and $f:X\to X$ be one-to-one and continuous. Is $f$ surjective?
2answers
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### Question about finite sets/compactness

I understand that every finite subset of a metric space is compact. But are there any topological spaces where finite sets are not compact? Is that even possible? I don't think it is but I just want ...
1answer
43 views

### On compact topological group

Must a compact topological group be metrizable? If not, is it separable? Thanks for any help.
10answers
998 views

### How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
2answers
32 views

### Is every countably compact space feebly compact?

A topological space is said to be feebly compact if every locally finite cover by nonempty open sets is finite. Every compact space is feebly compact but how about countably compact spaces?
1answer
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### Under what condition only does every compact subset of $X$ is closed implies $X$ Hausdorff?

It is trivial to see that: If $X$ is Hausdorff, then every compact subset of $X$ is closed. I am asking under what condition does the converse hold, i.e. when does If every compact subset of $X$ is ...
2answers
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### Hausdorff space and Cantor's intersection theorem

$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact. I tried to prove by ...
2answers
56 views

### one point compactification

I am asked to describe the one point compactification of $(0,1) \cup [2,3)$ of $\Bbb R$ and if I'm not mistaken it is just a circle union the closed set [2,3] correct? Am I missing something?
0answers
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### Another question in relation to Tychonoff theorem

Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
1answer
91 views

### How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?

A topological space $X$ is called sequentially compact if every sequence of points in $X$ has a subsequence that converges to a point in $X$. I know it's very similar to Bolzano–Weierstrass theorem ...
2answers
90 views

### Compact space and Hausdorff space

A continuous map from a compact space to a Hausdorff space is closed. Why this is true? Help me please I want to learn why this is correct.
1answer
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### Separating disjoint compact sets in Hausdorff space by using open sets

If $X$ is a Hausdorff Space and there are 2 disjoint compact sets $A,B\subset X$, we want to prove $\exists V,U \subset X, V \bigcap U=\emptyset$ S.T. $A\subset U, B\subset V$.($U,V$ are open sets) ...
1answer
35 views

### A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists$ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
1answer
54 views

### Compactness and connectedness on $M_n(\mathbb R)$

Consider $M_n(\mathbb R)$, the set of all $n\times n$ matrices. Which of the following are compact and which are connected? a) The set of all invertible matrices b) The set of all orthogonal ...
6answers
284 views

### Give an example of a simply ordered set without the least upper bound property.

In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact." (The LUB ...
1answer
111 views

### Does there exist a topology for a set $X$ which is compact and Hausdorff?

For every set $X$ and every topology $\tau$ over $X$ we have that $\tau$ contains the trivial topology $\{ X, \emptyset\}$, which is compact, and is contained in the discrete topology \$\{ S: S ...
1answer
42 views

### Pointwise Sup of continuous functions using Compactness

Consider two topological spaces $X$, $Y$. Assume $X$ is compact. Question: Is it true that for any continuous function $f:X\times Y\rightarrow Z$ (where $X\times Y$ has the product topology and ...
1answer
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### Compactification of a discrete space using ultrafilters.

I want to show for the collection of ultra filters on a (non-empty) set $A$, $U(A)$. That $U(A)$ is compact where the topology is derived from the base $U_B = \{F\in U(A)|B\in F\}$. Seeing as $A$ can ...
1answer
34 views

### Metacompactness of the Euclidean space

Does anyone know how to prove that every Euclidean space is countably metacompact? In particular, my interest is in $R^2$. Thanks, Shir
1answer
54 views

### How to show that a continuous map on a compact metric space must fix some non-empty set.

Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$ I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
1answer
109 views

### Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology?

My question is: Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology? How do you prove this? Do you use the countable open covering or do you use the accumulation point ...
1answer
69 views

### Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
1answer
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### Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.

I would like to show that a completely regular topological space is locally compact iff it is (weak-star) open in its Stone-Čech compactification. Does this hold in general? I.e given a compact ...
1answer
44 views

### How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
1answer
56 views

### Is a minimal Hausdorff uniformity compact?

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$ Is $(X,\mathcal D)$ compact?
1answer
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### equivalence of compactness and countably compactness

Is there a way to prove that in metric spaces, compactness and countably compactness are equivalent, without using the Bolzano Weierstrass Property?
1answer
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### exercises in compactness

I am working on some practice problems on Compactness. (Q.1.a Chapter 1.7 in Advanced Calculus, Folland) The question is : Give an example of : a closed set $S\subset R\quad$ and a continuous ...
1answer
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### No unbounded real continuous function on $X$ can be extended to a continuous real function on $\beta X$

By the Čech-Stone compactification theorem, I know that if $X$ is Tychonoff and $f:X\to [a,b]$ is continuous then $f$ can be extended to $\hat{f}:\beta X\to [a,b]$. How can we show that no unbounded ...
1answer
42 views

### Show $\tau=\tau^*$ if $\tau^*\subset \tau$ [duplicate]

Let $(X,\tau)$ be compact and $(X,\tau^*)$ be a Hausdorff space. How can we show that $\tau=\tau^*$ if $\tau^*\subset \tau$?
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