Compactness
Compactness [ edit ] Main article: Compact space Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space. There are several equivalent definitions of compactness in metric spaces: A metric space M is compact if every open cover has a finite subcover (the usual topological definition). A metric space M is compact if every sequence has a convergent subsequence. (For general topological spaces this is called sequential compactness and is not equivalent to compactness.) A metric space M is compact if it is complete and totally bounded. (This definition is written in terms of metric properties and doesn't make sense for a general topological space, but it is nevertheless topologically invariant since it is equivalent to compactness.) One example of a compact space is the closed interval [0, 1] . Compactness is important for similar reasons to completeness: it makes it ...