## Proof That a Topological Space With the Cofinite Topology is Compact

Theorem

A topological spacewith the cofinite topologyis compact.

Proof

Supposeis an open cover ofChoose

Sinceis the cofinite topologyis a finite set so we can write

is a cover ofso for each of the elements inat least oneexists such thatfor

Henceand andis compact.