\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

Compactness. Examples

• The collection of sets \( \{(n - 1, n + 1) : n \in \mathbb{Z} \} \) form an open cover of \( \mathbb{R} \). This cover has [no finite subcover/a finite subcover?].
• Divide the interval \( [0, 1] \) like so: \( U_0 = [0, \frac{1}{2}] \), \( U_n = (2^{-n}, 1] \) for \( n \ge 1 \), then form the cover \( (U_n)_{n \ge 0} \). This is an open cover, and has many finite subcovers (e.g. ([0, 1/2), (1/2, 1], (1/4, 1]). So we can say that \( [0, 1] \) is [...].
• The compact subspaces of \( \mathbb{R}^n \) are the [...].
• Any indiscrete space is [compact/not compact, possibly with conditions].
• Any finite space is [compact/not compact, possibly with conditions].
• A discrete space is [compact/not compact, possibly with conditions].
• In a normed vector space \( V \), the closed unit ball is [compact/not compact, possibly with conditions].