Show that any infinite set with the cofinite topology is compact.
Solution
1. In the cofinite topology, every nonempty open set has finite complement.
This is the definition of the cofinite topology.
2. Let \(\mathcal U\) be an open cover and choose one nonempty \(U_0\in\mathcal U\). Then \(X\setminus U_0\) is finite.
Apply 1.
3. For each of the finitely many points of \(X\setminus U_0\), choose one member of \(\mathcal U\) containing it. Together with \(U_0\) these form a finite subcover.
Every point is covered because \(\mathcal U\) is a cover.
4. Hence every space with the cofinite topology is compact.
Every open cover has the finite subcover constructed in 3.