Compact Space

Compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space

A subset of Euclidean space in particular is called compact if it is closed and bounded, which implies that any infinite sequence from the set has a subsequence that converges to a point in the set,

Formally, a topological space $X$ is called compact if every open cover of $X$ has a finite sub-cover. That is, $X$ is compact if for every collection $C$ of open subsets of $X$ such that $X={\bigcup }_{S\in C}S$, there is a finite sub-collection $F\subset C$ such that $X={\bigcup }_{S\in F}S$