Compactness of Topological Space

Cover and Subcover

A cover of a set $A$ is collection $\mathcal{U}$ of sets whose union contains $A$: $A\subset {\bigcup }_{U\in \mathcal{U}}U$ A subcover of a cover $\mathcal{U}$ is a subset of $\mathcal{U}$ whose elements still cover $A$. A cover is open if all its elements are open.

e.g. $\{(n,n+3)\mid n\in \mathbb{Z}\}$ is an open cover of $\mathbb{R}$; $\{(2k,2k+3)\mid k\in \mathbb{Z}\}$ is a subcover.

Compactness

A topological space $T$ is compact if every open cover of $T$ has a finite subcover. A subset $S$ of $T$ is compact if every open cover of $S$ by subsets of $T$ has a finite subcover. This is the same as $S$ being compact with the subspace topology.

e.g. $(0,1)$ is not compact, $\{(0,a)\mid a\in (0,1)\}$ is an open cover with no finite subcover; $\mathbb{R}$ is not compact, $\{(-\infty ,a):a\in \mathbb{Z}\}$ has no finite subcover.

Sequential Compactness

Let $X$ be a topological space and $A\subset X$. We say that $A$ is sequentially compact if every sequence in $A$ has a subsequence converges to a point in $A$.

Lemma

If $T$ is a topological space and $S\subset T$ then $S$ is compact in the $T$ if and only if $(S,{\mathcal{T}}_S)$ is compact.

Theorem

Let $X$ be a compact topological space and $K$ a closed subspace of $X$. Then $K$ is compact.

Proof Let $K\subset X$ be closed and let $\{U_{\alpha }{\}}_{\alpha \in \Lambda }$ be an open covering of $K$. Then $\{K^c\cup U_{\alpha }{\}}_{\alpha \in \Lambda }$ is an open covering of $X$. Since $X$ is compact, there exist ${\alpha }_1,\dots ,{\alpha }_n\in \Lambda$ such that $X=K^c\bigcup \left({\bigcup }_{i=1}^n{U}_{{\alpha }_i}\right)$It follows that $K\subset {\bigcup }_{i=1}^n{U}_{{\alpha }_i}$, therefore $K$ is compact. $\square$

Proposition

Any compact subset $K$ of a Hausdorff space $T$ is closed.

Tikhonov Theorem

Def Product of Sets Let $\{X_{\alpha }{\}}_{\alpha \in A}$ be a family of sets. We define ${\prod }_{\alpha \in A}{X}_{\alpha }$ to be the collection of all functions $x:A\to {\bigcup }_{\alpha \in A}{X}_{\alpha }$ such that $x(\alpha )\in X_{\alpha }$ for each $\alpha \in A$.

Def Projection

References and Other Resources

• Morphocular, The Concept So Much of Modern Math is Built On