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.

You have no rights to post comments