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 open sets , respectively such that
Theorem
The followings are equivalent:
A topological space is ;
Every singleton set is closed for all ;
For , the intersection of all open sets containing , is .
Proof If is , for any singleton set , there are no limit points, so the closure of is , which is closed; Now assume every singleton set is closed, then for any point in the intersection of all open sets containing , if , since is closed, is an open set containing , but which is a contradiction; Lastly, suppose the intersection of all open sets containing is , and , are two distinct points, then and serve as open sets containing and respectively, but not the other.
Hausdorff Space
Hausdorff Space
A topological space is called Hausdorff or 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 .