Separation and Hausdorff Spaces

The ​ hierarchy is a way of classifying separation axioms in topology. They describe how well points and sets can be distinguished by open sets.

Kolmogorov Space

Kolmogorov Space

A topological space is called a Kolmogorov space or space if for any two distinct points in the space, there exists an open set that contains one of the points but not the other.

Fréchet Space

Space

A topological space is called Fréchet or space if for any two distinct points in the space, there exists an open set that contains one of the points but not the other, and vice versa.

Theorem

A topological space is a Fréchet space if and only if every singleton set is closed for all .

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 .