Show that \(\RR\) with the cofinite topology is compact.
Exercise
Solution
1. Let \(\mathcal U\) be an open cover of \(\mathbb R\) in the cofinite topology and choose a nonempty \(U_0\in\mathcal U\).
A cover of the nonempty space contains a nonempty member.
2. The complement \(\mathbb R\setminus U_0\) is finite.
This is the definition of a nonempty cofinite-open set.
3. Choose one member of \(\mathcal U\) for each point of this finite complement. Together with \(U_0\) they form a finite subcover.
\(U_0\) covers everything except finitely many points, and the chosen additional sets cover those points.
4. Hence \(\mathbb R\) with the cofinite topology is compact.
Every open cover has a finite subcover by <1>1–<1>3.