Question

Prove that any set with the cofinite topology is compact

Answer 1

Let us prove that any set $A$ with the cofinite topology is compact. Let $\mathcal U$ be an open cover of the set $A.$ Let $U_0$ is an element of $\mathcal U.$ It follows from definition of cofinite topology that $A\setminus U_0$ is a finite set. Let $A\setminus U_0=\{x_1,\ldots, x_n\}.$ Since $\mathcal U$ is a cover, for each $k\in\{1,\ldots,n\}$ there are exists $U_k\in\mathcal U$ such that $x_k\in U_k.$ Consequently, $U_0\cup U_1\cup\ldots\cup U_n\supset A$ and hence, $\{U_0,U_1,\ldots, U_n\}\subset\mathcal U$ is a finite subcover of $A.$ We conclude that $A$ is compact.