mathematics, a Hausdorff space (/ˈhaʊsdɔːrf/ HOWSS-dorf, /ˈhaʊzdɔːrf/ HOWZ-dorf), T2 space or separated space, is a topological space where distinct points...

normal space is a topological space in which any two disjoint closed sets have disjoint open neighborhoods. Such spaces need not be Hausdorff in general...

topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which...

neighborhoods. A T3 space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological...

Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it...

compact space into a Hausdorff space is a homeomorphism. A compact Hausdorff space is normal and regular. If a space X is compact and Hausdorff, then no...

completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff). Paul Urysohn had...

Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others do...

space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space,...

mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization...

particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense...

to as the Hausdorff–Besicovitch dimension. More specifically, the Hausdorff dimension is a dimensional number associated with a metric space, i.e. a set...

a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the...

mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each...

a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This...

For example, the line with two origins is not a Hausdorff space but is locally Hausdorff. Sierpiński space is a simple example of a topology that is T0 but...

neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint...

space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise. With this topology, every Banach space is a Baire space,...

denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension...

onto the unit square (any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism). But a unit square has no cut-point, and...

axioms for accessible spaces, Hausdorff spaces, regular spaces, and normal spaces. Topologists assigned these classes of spaces the names T1, T2, T3,...

totally disconnected Hausdorff space that does not have small inductive dimension 0. Extremally disconnected Hausdorff spaces Stone spaces The Knaster–Kuratowski...

Felix Hausdorff (/ˈhaʊsdɔːrf/ HOWS-dorf, /ˈhaʊzdɔːrf/ HOWZ-dorf; November 8, 1868 – January 26, 1942) was a German mathematician, pseudonym Paul Mongré...

compact metric space (Dunford & Schwartz 1958, p. 382). Modern formulations of the theorem allow for the domain to be compact Hausdorff and for the range...

above agree for separable, metrisable spaces.[citation needed][clarification needed] A zero-dimensional Hausdorff space is necessarily totally disconnected...

with metric spaces, every uniform space X {\displaystyle X} has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space Y {\displaystyle...

Completely normal Hausdorff A completely normal Hausdorff space (or T5 space) is a completely normal T1 space. (A completely normal space is Hausdorff if and only...

Hausdorff or locally connected. A space in which all components are one-point sets is called totally disconnected. Related to this property, a space X...

Every R1 space is also R0. X is Hausdorff, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and...

space to be metrizable. Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and...