Find a concept
Start typing to search the mathematical index.
In a Hausdorff space, every compact set is closed
Compactness implies closedness: Let be a Hausdorff space and let be compact. Then is closed in .
In particular, this applies to subsets of any metric space, since metric spaces are Hausdorff.
A Hausdorff space (or T2 space) is a topological space such that for any distinct points there exist neighborhoods of and of with .
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A closed set in a topological space is a subset such that its complement (the set difference) is a open set.
Closed sets are the natural targets of the closure operation: the closure of is the smallest closed set containing . Closed sets also provide an equivalent formulation of continuity via preimages of closed sets.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.