A topological space is compact if every open cover of 
has a finite subcover. In other words, if 
is the union of a family of open sets,
there is a finite subfamily whose union is 
. A subset 
of a topological space 
is compact if it is compact as a topological space with the relative
topology (i.e., every family of open sets of 
whose union contains 
has a finite subfamily whose union contains 
).
Compact Space
See also
Compact Set, Heine-Borel Theorem, Paracompact Space, Topological SpaceExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Compact Space." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompactSpace.html