Compactness Review

Table of Contents

We will now review some of the recent material regarding compactness in topological spaces.

Recall from the Covers of Sets in a Topological Space page that if $$X$$ is a topological space then an a Cover of $$X$$ is a collection of sets $\mathcal F$ such that:

(1)

\begin{align} \quad X \subseteq \bigcup_{U \in \mathcal F} U \end{align}

If the sets in $\mathcal F$ are all open (or closed) and satisfy the inclusion above then $\mathcal F$ is called an Open Cover (or Closed Cover) of $$X$$ .

On the Compactness of Sets in a Topological Space page we said that a set $A \subseteq X$ is said to be Compact in $$X$$ if every open cover of $$A$$ has a finite subcover. We saw that the set of natural numbers $\mathbb{N}$ is NOT compact in $\mathbb{R}$ with the usual topology.

On the Compactness of Finite Sets in a Topological Space we saw a very nice theorem regarding the compactness of finite sets. We saw that if $A = \{ x_1, x_2, ..., x_n \}$ is a finite subset of a topological space $$A$$ is compact in $$X$$ . This is because if we are given an open cover $\mathcal F$ of $$A$$ then all we need to do is select a set in the open cover associated to each point $x_i \in A$ , and so, we will always be able to construct a finite subcover of size less than or equal to $$n$$ .

On the Finite Intersection Property Criterion for Compactness in a Topological Space page we looked at a nice result with regards to compactness of a topological space.

First, if $\mathcal F$ is any collection of sets then we said that $\mathcal F$ has the Finite Intersection Property if every finite collection of sets $\{ F_1, F_2, ..., F_n \} \subseteq \mathcal F$ we have that:

(2)

\begin{align} \quad \bigcap_{i=1}^{n} F_i \neq \emptyset \end{align}

In other words, the intersection of all sets in a finite subset of $\mathcal F$ is nonempty.

We then proved that a topological space $$X$$ is compact if and only if for every collection of closed sets $\mathcal F$ of $$X$$ that have the finite intersection property then the intersection of all sets in $\mathcal F$ is nonempty, i.e. $\displaystyle{\bigcap_{F \in \mathcal F} F \neq \emptyset}$ .

On the Preservation of Compactness under Continuous Maps we saw that if $f : X \to Y$ is a continuous map and $$X$$ is a compact topological space then the range $$f(X)$$ is also a compact space.

On the Closed Sets in Compact Topological Spaces we saw that if $$X$$ is a compact topological space, $A \subseteq X$ , and $$A$$ is closed in $$X$$ then $$A$$ is also compact in $$X$$ .

On the Compact Sets in Hausdorff Topological Spaces page we proved that if $$X$$ is a Hausdorff topological space (i.e., for every pair of distinct points $x, y \in X$ there exists open neighbourhoods $$U$$ of $$x$$ and $$V$$ of $$y$$ such that $U \cap V = \emptyset$ ) then every compact set $A \subseteq X$ is closed.

We then began to see how useful the concept of a compact topological space was. On the Homeomorphisms Between Compact and Hausdorff Spaces page we saw that if $f : X \to Y$ , $$X$$ is compact, $$Y$$ is Hausdorff, and $$f$$ is a continuous bijective function, then $$f$$ is necessarily a homeomorphism between $$X$$ and $$Y$$ .

Furthermore, on the Quotients from Equivalence Relations defined by Functions from Compact to Hausdorff Spaces we first proved that if $f : X \to Y$ , $$X$$ is compact, $$Y$$ is Hausdorff, and $$f$$ is a continuous map then $$f$$ is also a closed map (i.e., the image of any closed set in $$X$$ is closed in $$Y$$ ).

We then proved a nice result. We proved that if $f : X \to Y$ , $$X$$ is compact, $$Y$$ is Hausdorff, and $f : X \to Y$ is continuous and surjective then for the equivalence relation $\sim_f$ defined for all $x_1, x_2 \in X$ by $x_1 \sim_f x_2$ if and only if $$f(x_1) = f(x_2)$$ , we have that the quotient space $X /\: \sim_f$ is homeomorphic to $$Y$$ .

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