Compact Space

A Topological Space is compact if every open cover of $X$ has a finite subcover. In other words, if $X$ is the union of a family of open sets, there is a finite subfamily whose union is $X$. A subset $A$ of a Topological Space $X$ is compact if it is compact as a Topological Space with the relative topology (i.e., every family of open sets of $X$ whose union contains $A$ has a finite subfamily whose union contains $A$).