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Section 24.1 Compactness
Objectives
Work with open covers and finite subcovers to understand compactness.
Use compactness to guarantee boundedness and attainment of extrema.
Connect compactness, subsequences, and convergence through the Heine-Borel and Bolzano-Weierstrass theorems.
An open cover of \(A\) is a collection \(\set{U_\alpha}_{\alpha\subseteq S}\) such that
\begin{equation*}
A\subseteq\bigcup_{\alpha\in S}U_{\alpha}\text{.}
\end{equation*}
If
\(\set{U_\alpha}_{\alpha\subseteq S}\) has a finite subset
\(\set{U_\alpha}_{\alpha\subseteq F}\) (with
\(F\subseteq S\) ) which is still a cover of
\(A\text{,}\) then
\(\set{U_\alpha}_{\alpha\subseteq F}\) is called a finite subcover of
\(A\text{.}\)
A set
\(A\) is compact if every open cover of
\(A\) admits a finite subcover of
\(A\text{.}\) Great note I saw online: “A has B” suggests that B is part of the nature of A. For example, every Cauchy sequence (in
\(\bb R\) ) has a limit. “A admits B” suggests that it is possible to find B, but this existence is not inherent to A, and we may not be able to find B explicitly.
Heine-Borel theorem: A set
\(S\subseteq\bb R^{n}\) is compact if and only if
\(S\) is closed and bounded.
A continuous function on a compact set is bounded.
Extreme value theorem: A continuous function on a compact set attains its supremum and infimum.
A sequence
\((a_{n})\) is monotone increasing if
\(a_{n}\le a_{n+1}\) for all
\(n\text{.}\) Similarly,
\((a_{n})\) is monotone decreasing if
\(a_{n}\ge a_{n+1}\) for all
\(n\text{.}\)
Monotone convergence theorem: A monotone sequence converges if and only if it is bounded. Moreover, if
\((a_{n})\) is increasing, then
\(\limit{n\to\infty}a_{n}=\sup\set{a_n:n\in\bb N}\text{,}\) and if
\((a_{n})\) is decreasing, then
\(\limit{n\to\infty}a_{n}~=\inf\set{a_n:n\in\bb N}\)
Bounded sets contain bounded sequences.
If \(n_{1}<n_{2}<n_{3}<\cdots\text{,}\) then
\begin{equation*}
a_{n_1},a_{n_2},a_{n_3},\cdots
\end{equation*}
is called a subsequence of \((a_{n})\) and is denoted \((a_{n_k})\text{.}\)
A sequence converges to
\(a\) if and only if every subsequence converges to
\(a\text{.}\)
If
\((a_{n})\) has two subsequences that converge to different limits, then
\((a_{n})\) diverges.
If
\((a_{n})\) is monotone and has a subsequence converging to
\(a\text{,}\) then
\((a_{n})\) also converges to
\(a\text{.}\)
Bolzano-Weierstrass theorem: Every bounded sequence has a convergent subsequence.
A space is sequentially compact if every sequence has a convergent subsequence.
