Let \(X\) be the set of integers, let \[\mathcal C_1:=\{A\subseteq X\mid X\setminus A\text{ is finite}\},\qquad \mathcal C_2:=\{A\subseteq X\mid 0\notin A\}.\] Show that the union \(\mathcal T:=\mathcal C_1\cup\mathcal C_2\) is a topology on \(X\), and show that this topological space is compact.
Solution
1. \(\mathcal T\) is a topology on \(X\).
1.1. \(\varnothing,X\in\mathcal T\).
\(0\notin\varnothing\), so \(\varnothing\in\mathcal C_2\); and \(X\setminus X=\varnothing\) is finite, so \(X\in\mathcal C_1\).
1.2. \(\mathcal T\) is closed under arbitrary unions.
Let \(U=\bigcup_{\alpha}U_\alpha\) with each \(U_\alpha\in\mathcal T\). If some \(U_{\alpha_0}\in\mathcal C_1\), then \(X\setminus U\subseteq X\setminus U_{\alpha_0}\) is finite, so \(U\in\mathcal C_1\). Otherwise every \(U_\alpha\in\mathcal C_2\), so \(0\notin U\) and \(U\in\mathcal C_2\).
1.3. \(\mathcal T\) is closed under finite intersections.
Let \(U,V\in\mathcal T\). If \(U,V\in\mathcal C_1\), then \(X\setminus(U\cap V)=(X\setminus U)\cup(X\setminus V)\) is finite, so \(U\cap V\in\mathcal C_1\). Otherwise one of them, say \(U\), lies in \(\mathcal C_2\), so \(0\notin U\cap V\) and \(U\cap V\in\mathcal C_2\).
1.4. Q.E.D.
Steps 1, 2, and 3 are the axioms of a topology.
2. \((X,\mathcal T)\) is compact.
Let \(\mathcal U\subseteq\mathcal T\) cover \(X\), and choose \(U_0\in\mathcal U\) with \(0\in U_0\). Then \(U_0\notin\mathcal C_2\), so \(U_0\in\mathcal C_1\) and \(X\setminus U_0=\{x_1,\ldots,x_k\}\) is finite. Choose \(U_i\in\mathcal U\) with \(x_i\in U_i\) for \(1\le i\le k\); then \(\{U_0,U_1,\ldots,U_k\}\) covers \(X\).
3. Q.E.D.
Steps 1 and 2 prove both assertions.