Hausdorff Space

Hausdorff Space

A topological space is called Hausdorff if for any with there exist open neighborhoods of and of such that .

Theorem

In Hausdorff spaces, limits of sequences are unique if they exist.

Proof Assume a sequence in a Hausdorff space has two distinct limits and . Then there exist neighborhoods of and of such that . However, since and , there exists an integer such that and for . Thus , which is a contradiction.

Theorem

Every finite set in a Hausdorff topological space is closed.

Proof It suffices to show for any the set is closed. For any , by the Hausdorff property we can find an open set containing but . Thus and hence it is open. Consequently is closed.

Theorem

Let be a Hausdorff space. If is a finer topology on , then is also a Hausdorff space.

Proof Let with . Since , the open sets in are also open in . Thus there exist open sets such that , and , that is is a Hausdorff space under .

Thrm Let be a Hausdorff space and . A point is a limit point of if and only if any neighborhood of contains infinitely many points of .