Proof That a Closed Subset of a Countably Compact Space is Countably Compact
Theorem
A closed subset
of a countably compact space
is countably compact.
Proof
Letbe a countably compact set and letbe a closed subset of
Let
be an infinite subset of
A is also an infinite subset of a countably compact space
An accumulation point
ofexists such that
Since
is also an accumulation point ofSinceis closed, it contains all it's accumulation points, so
Hence an infinite subsetof a closed subsethas an accumulation point
henceis countably compact.
