Theorem
The interval
is 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 subsets
of
Let
This set is an open cover ofbecause
so inductively
and
Hence no finite subcover exists andis not compact.