Definition 6.1.
A collection \(\mc A\subseteq \mc P(X)\) is said to cover a subset \(Y\subseteq X\) iff \(Y\subseteq\bigcup\mc A\text{.}\)
Section 6 Compactness
Definition 6.2.
A subset \(K\subseteq X\) of a topological space is said to be compact iff for every collection \(\mc U\) of open sets covering \(K\text{,}\) there exists a finite subcollection \(\mc F\subseteq\mc U\) that also covers \(K\text{.}\)
Checkpoint 6.3.
Determine if each of the following subsets of the Euclidean line \(\mb R\) is compact.
- \(\displaystyle \mb R\)
- \(\displaystyle \mb Z\)
- \(\displaystyle \setBuilder{2^{-n}}{n\in\omega}\)
- \(\displaystyle \setList{0}\cup\setBuilder{2^{-n}}{n\in\omega}\)
- \(\displaystyle (0,1)\)
Definition 6.4.
A subset \(A\subseteq X\) of a space with metric \(d(x,y)\) is said to be bounded iff there exists some \(D\in[0,\infty)\) with \(d(x,y)<D\) for all \(x,y\in A\text{.}\)
Theorem 6.5.
Every compact subset of a Hausdorff space is closed.
Proposition 6.6.
Let \(\mc T=\setList{\emptyset}\cup
\setBuilder{\omega\setminus F}{F\text{ is finite}}\) be the cofinite topology on \(\omega\text{.}\) Every subset of \(\omega\) is compact under this topology.
Theorem 6.7.
For every subset \(K\subseteq\mathbb R^n\) with the Euclidean metric, \(K\) is compact if and only if it is closed and bounded.
Definition 6.8.
A subset \(R\subseteq X\) of a topological space is said to be relatively compact iff for every collection of open sets \(\mc U\) covering \(X\text{,}\) there exists a finite subcollection \(\mc F\subseteq\mc U\) that covers \(K\text{.}\)
Theorem 6.9.
Let \(X\) be regular. A subset \(R\subseteq X\) is relatively compact if and only if \(\cl R\) is compact.
Corollary 6.10.
For every subset \(K\subseteq\mathbb R^n\) with the Euclidean topology, \(K\) is relatively compact if and only if it is bounded.
Theorem 6.11.
Let \(X\) be compact and \(K\) be a closed subset of \(X\text{.}\) Then \(K\) is compact.
Proposition 6.12.
Every finite subset of a space is compact.
Proposition 6.13.
Every finite union of compact subsets is compact.
Theorem 6.14.
Let \(f:X\to Y\) be continuous and \(K\subseteq X\) be compact. Then \(f[K]\) is compact.
Corollary 6.15.
Compactness is a topological property.
Theorem 6.16.
Every infinite subset of a compact set has a limit point.
Theorem 6.17.
Let \(\mc K=\setBuilder{K_n}{n\in\omega}\) be a collection of non-empty compact subsets of a topological space such that \(K_{n+1}\subseteq K_n\) for all \(n\in\omega\text{.}\) Then \(\bigcap\mc K\) is a non-empty compact set.
Theorem 6.18.
Let \(X\) be metrizable. Then the following are equivalent for \(K\subseteq X\text{.}\)
- \(K\) is compact
- Every infinite subset of \(K\) has a limit point.
- Every infinite subset of \(K\) has a sequential limit point.
Lemma 6.19.
A topological space \(X\) is Hausdorff if and only if for every pair of disjoint compact subsets \(H,K\) there exist disjoint open sets \(U,V\) such that \(H\subseteq U\) and \(K\subseteq V\text{.}\)
Theorem 6.20.
Every compact Hausdorff space is \(T_4\text{.}\)
