# The Unapologetic Mathematician

## Some compact subspaces

Let’s say we have a compact space $X$. A subset $C\subseteq X$ may not be itself compact, but there’s one useful case in which it will be. If $C$ is closed, then $C$ is compact.

Let’s take an open cover $\{F_i\}_{i\in\mathcal{I}}$ of $C$. The sets $F_i$ are open subsets of $C$, but they may not be open as subsets of $X$. But by the definition of the subspace topology, each one must be the intersection of $C$ with an open subset of $X$. Let’s just say that each $F_i$ is an open subset of $X$ to begin with.

Now, we have one more open set floating around. The complement of $C$ is open, since $C$ is closed! So between the collection $\{F_i\}$ and the extra set $X\setminus C$ we’ve got an open cover of $X$. By compactness of $X$, this open cover has a finite subcover. We can throw out $X\setminus C$ from the subcover if it’s in there, and we’re left with a finite open cover of $C$, and so $C$ is compact.

In fact, if we restrict to Hausdorff spaces, $C$ must be closed to be compact. Indeed, we proved that if $C$ is compact and $X$ is Hausdorff then any point $x\in X\setminus C$ can be separated from $C$ by a neighborhood $U\subseteq X\setminus C$. Since there is such an open neighborhood, $x$ must be an interior point of $X\setminus C$. And since $x$ was arbitrary, every point of $X\setminus C$ is an interior point, and so $X\setminus C$ must be open.

Putting these two sides together, we can see that if $X$ is compact Hausdorff, then a subset $C\subseteq X$ is compact exactly when it’s closed.

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Posted by John Armstrong | Point-Set Topology, Topology

## 3 Comments	»

1. [...] Heine-Borel Theorem We’ve talked about compact subspaces, particularly of compact spaces and Hausdorff spaces (and, of course, compact Hausdorff spaces). So [...]

Pingback by | January 18, 2008 | Reply

2. [...] is a closed interval covered by the open intervals . But the Heine-Borel theorem says that is compact, and so we can find a finite collection of the which cover . Renumbering the open intervals, we [...]

Pingback by | April 15, 2010 | Reply

3. [...] base since it’s open. This union gives an open covering of the set. Since the set is closed, it is compact. And so the open covering we just found has a finite subcover. That is, we can write our clopen set [...]

Pingback by | August 19, 2010 | Reply

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## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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