## Proof That Every Closed and Bounded Interval With the Absolute Value Topology is Compact

Theorem

Every closed and bounded intervalofwith the absolute value topology is compact.

Proof

The proof is done with the aid of the Heine - Borel Theorem:

Letbe a closed and interval inand letbe a collection of sets open in satisfyingthen we can select a finite number of thesaysatisfying

Hence every open coverofhas a finite subcoverandis compact.