“A piece of art is a compact form of the universe”
―Thomas Kinkade
Motivation: Characterize spaces for which the Extreme Value Theorem holds.
Definition3.4.1.
Let \(X\) be a topological space. A collection \(\cC\) of subsets of \(X\) is said to cover \(X\text{,}\) or to be a covering of \(X\text{,}\) if \(\cup C \in \cC C = X\text{.}\) An open covering of a topological space \(X\) is a collection \(\cC\) of open sets in \(X\) that covers \(X\text{.}\)
Definition3.4.2.
A topological space \(X\) is compact if every open covering of \(X\) contains a finite subcollection that also covers \(X\text{.}\)
Theorem3.4.3.EVT = Extreme Value Theorem.
Let \(X\) be a compact space, and let \(f: X \to (\R,\cT_{\text{Eucl}})\) be a continuous function. Then there exist \(c,d \in X\) such that for all \(p \in X\text{,}\)\(f(c) \leq f(p) \leq f(d)\text{.}\)
Example3.4.4.
Examples
Interactions with constructions and continuous functions:.
Theorem3.4.5.
Let \(A\) be a subspace of a topological space \((X,\cT_X)\text{.}\) The space \(A\) is compact if and only if for every collection \(\cC\) of open sets in \(X\) satisfying \(A \sse \cup C \in \cC C\text{,}\) there is a finite subcollection \(\cD \sse \cC\) such that \(A \sse \cup D \in \cD D\text{.}\)
Theorem3.4.6.A continuous image of a compact space is compact.
If \(X,Y\) are topological spaces, and if \(X\) is compact and \(f: X \to Y\) is a continuous surjective function, then \(Y\) is compact.
Corollary3.4.7.
Compactness is a homeomorphism invariant.
Corollary3.4.8.
If \(X\) is a compact space and \(X/\sim\) is a quotient space, then \(X/\sim\) is compact.
Theorem3.4.9.
If \(A\) is a subspace of a compact space \(X\) and \(A\) is a closed subset in \(X\text{,}\) then \(A\) is compact.
Theorem3.4.10.Every compact subspace of a Hausdorff space is closed.
If \(Y\) is a compact subspace of a \(T_2\) space \(X\text{,}\) then \(Y\) is a closed subset of \(X\text{.}\)
Corollary3.4.11.
If \(Y\) is a compact subspace of a \(T_2\) space \(X\text{,}\) and \(p \in X\setminus Y\text{,}\) then there are disjoint open sets \(U,V\) in \(X\) satisfying \(p \in U\) and \(Y \sse V\text{.}\)
Theorem3.4.12.
If \(X\) and \(Y\) are compact spaces, then the product space \(X \times Y\) is compact. (a') ("Tychonoff theorem") If \(X_\alpha\) is a compact space for all \(\alpha\text{,}\) then the product space \(\prod_\alpha X_\alpha\) is compact.
Tube Lemma.
If \(X\) and \(Y\) are spaces, \(p \in X, Y\) is compact, and \(N\) is an open set in the product space \(X \times Y containing\{p\} \times Y\text{,}\) then there is an open set \(W\) in \(X\) such that\(\{p\} \times Y \sse W \times Y \sse N\text{.}\)
Remark3.4.13.
Compactness is not preserved by non-closed subspaces or continuous preimages.
Theorem3.4.14.VUT = Very Useful Theorem.
A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
Corollary3.4.15.CH-HBT = Compact Hausdorff Homeomorphism Building Theorem.
Let \(X\) be a compact space, let \(\sim \) be an equivalence relation on \(X\text{,}\) and let \(X/\sim\) be the quotient space. If \(Z\) is a \(T_2\) space and \(f: X \to Z\) is a continuous surjection satisfying \(f(p) = f(q)\) if and only if \(p \sim q\) for all \(p,q \in X\text{,}\) then the function \(g: X/\sim \to Z\) defined by \(g([p]) := f(p)\) for all \([p] \in X/\sim\) is a homeomorphism.
Theorem3.4.16.(CQ4).
If \(X\) is a compact space, \(Y\) is a \(T_2\) space, and \(f: X \to Y\) is a surjective continuous function, then \(f\) is a quotient map.
Compactness and Euclidean topology:.
Theorem3.4.17.
If \(a,b \in \R\) with \(a < b\text{,}\) then the interval \([a,b]\text{,}\) with the subspace topology from \((\R,\cT_{\text{Eucl}})\text{,}\) is compact.
Definition3.4.18.
Let \((X,d)\) be a metric space. The diameter of a nonempty subset \(Y\) of \(X\) is \(diam(Y) := sup{d(p,q) | p,q \in Y\}\text{.}\) The set \(Y\) is bounded if \(diam(Y)\) is finite.
Theorem3.4.19.Heine-Borel Theorem.
Let \(Y\) be a subspace of \((\R^n,\cT_{\text{Eucl}})\text{.}\) Then \(Y\) is compact iff \(Y\) is a closed and bounded subset of \(\R^n\text{.}\)
Proposition3.4.20.
If \(\sim\) is the smallest equivalence relation on \(I\) such that \(0 \sim 1\text{,}\) then \(I/\sim \cong S^1\text{,}\) with a homeomorphism that maps the equivalence class \([s]\) to the point \(( \cos(2 \pi s), \sin(2 \pi s) )\) for all \(s \in I\text{.}\)
If \(\sim\) is the smallest equivalence relation on \(I \times I\) such that \((s,1) \sim (1,1)\) for all \(s \in I\) and \((0,t) \sim (1,1) \sim (1,t)\) for all \(t \in I\text{,}\) then \((I \times I)/\sim \cong D^2\text{,}\) with a homeomorphism that maps the equivalence class \([(1,1)]\) to the point \((1,0)\) and maps \([(s,0)]\) to \((\cos(2\pi s),\sin(2\pi s))\) for all \(s \in I\text{.}\)
Theorem3.4.21.
Let \(X\) be a topological space, let \(\sim\) be an equivalence relation on \(X\text{,}\) and let \(q:X \to X/\sim\) be the quotient map. Let \(I = [0,1]\) have the Euclidean subspace topology.
The function \(q':X \times I \to (X/\sim ) \times I\) defined by \(q'(p,t) := (q(p),t)\) for all \((p,t) \in X \times I\) is a quotient map.
The relation \approx on X \times I defined by [(p,t) \approx (p',t') if and only if p \sim p' and t = t' for all (p,t),(p',t') \in X \times I] is an equivalence relation on X \times I.
The function h:(X \times I)/\approx \to (X/\sim ) \times I defined by h([(p,t)]) := [(q(p),t)] for all [(p,t)] \in (X \times I)/\approx is a homeomorphism.
Theorem3.4.22.Lebesgue Number Lemma.
If \(X\) is a compact metrizable space and \(\cC\) is an open covering of \(X\text{,}\) then there is a real number \(s > 0\) such that whenever \(A\) is a subset of \(X\) with diameter \(< s\text{,}\) then there is an open set \(U in \cC\) with \(A \sse U\text{.}\)
