A Countably Compact Set that is Not Compact
If a space is compact it is compact, but the converse is not true. A set may be countably compact but not compact.
Letbe the positive integers with topology consisting of the open setsand letbe a non empty subset of
LetIfis even thenis a limit point ofand ifis odd thenis a limit point of
Hencehas an accumulation point andis countably compact.
The setsconstitute an open cover ofand is not reducible to a finite subcover. Henceis not compact.