Hausdorff Space

It implies the uniqueness of limits of sequences nets and filters.

Hausdorff space. Any discrete space i e a topological space with the discrete topology is a hausdorff space. Let r d x y. In other words a topological space x is said to be a t 2 space or hausdorff space if for any x y x x y there exist open sets u and v such that x u y v and u v ϕ. They do not form a particularly nice category of spaces themselves but many such nice categories consist of only hausdorff spaces.

3 1a proposition every metric space is hausdorff in particular r n is hausdorff for n 1. Of the many separation axioms that can be imposed on a topological space the hausdorff condition is the most frequently used and discussed. In fact felix hausdorff s original definition of topological space actually required the space to be hausdorff hence the name. Definition a topological space x is hausdorff if for any x y x with x 6 y there exist open sets u containing x and v containing y such that u t v.

In mathematics the hausdorff distance or hausdorff metric also called pompeiu hausdorff distance measures how far two subsets of a metric space are from each other. It turns the set of non empty compact subsets of a metric space into a metric space in its own right. Hausdorff metric complete total boundedness ρ displaystyle rho connectivity reducible sets. The term metric space comes from hausdorff.

A hausdorff space is a topological space in which each pair of distinct points can be separated by a disjoint open set. It is named after felix hausdorff and dimitrie pompeiu. Hausdorff spaces are named after felix hausdorff one of the fou. There corresponds to each point at least one neighborhood and each neighborhood contains the point.

Hausdorff space definition a topological space in which each pair of points can be separated by two disjoint open sets containing the points. In topology and related branches of mathematics a hausdorff space separated space or t2 space is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. The axioms formulated by hausdorff 1919 for his concept of a topological space. Proof let x d be a metric space and let x y x with x 6 y.

Hausdorff spaces are a kind of nice topological space. Any two distinct points can be separated by disjoint open sets that is whenever p and q are distinct points of a set x there exist disjoint open sets up and uq such that up contains p and uq contains q. Typical examples euclidean space and more generally any manifold closed subset of euclidean space and any subset of euclidean space is hausdorff.