Proof That the Interval (0,1) is Not Compact With the Absolute Value Topology

Theorem

The intervalis not a compact space with the absolute value topology.

Proof

The open intervalwith the absolute value topology is Lindelof since it is a subspace of a second countable space setr . It is not compact however (if it were compact it would be Lindelof).

Consider the collection of open subsetsofLet

This set is an open cover ofbecause

so inductivelyand

Hence no finite subcover exists andis not compact.