Hausdorff Space

Any discrete space i e a topological space with the discrete topology is a hausdorff space.

Hausdorff space. The axioms formulated by hausdorff 1919 for his concept of a topological space. There corresponds to each point at least one neighborhood and each neighborhood contains the point. Let r d x y.

It implies the uniqueness of limits of sequences nets and filters. 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. It turns the set of non empty compact subsets of a metric space into a metric space in its own right.

Proof let x d be a metric space and let x y x with x 6 y. In fact felix hausdorff s original definition of topological space actually required the space to be hausdorff hence the name. A hausdorff space is a topological space in which each pair of distinct points can be separated by a disjoint open set.

They do not form a particularly nice category of spaces themselves but many such nice categories consist of only hausdorff spaces. Hausdorff space definition a topological space in which each pair of points can be separated by two disjoint open sets containing the points. Hausdorff spaces are a kind of nice topological space.

Hausdorff metric complete total boundedness ρ displaystyle rho connectivity reducible sets. These axioms describe the properties satisfied by subsets of elements in a neighborhood set of. 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.

A hausdorff space is a topological space with a separation property. The term metric space comes from hausdorff. Hausdorff spaces are named after felix hausdorff one of the fou.