Showing that $\mathbb{R}$ is compact with respect to the cofinite topology.

Question

We can define the cofinite topology as the following $\mathcal{T}_{cofinite} = \{U \subseteq X| U = \emptyset, X-U \mbox{is finite}\}$. Let $\mathcal{U}= \{U_{\alpha}\}_{\alpha \in I}$ be an open cover for $\mathbb{R}$. But where can we take this argument. We want to find a finite sub cover for $\mathbb{R}$ and then we are done. We can also included that since $\mathcal{U}= \{U_{\alpha}\}_{\alpha \in I}$ is an open cover for $\mathbb{R}$ that $\mathbb{R} = \cup_{\alpha \in I} U_{\alpha}$.

Answer 1

Take any (nonempty) set from your covering. It alone covers all $\mathbb{R}$ except maybe a finite number of elements. And a finite number of elements is obviously covered by some finite number of sets from your collection.