Hausdorff Space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.

Hausdorff spaces are named for Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom.

Read more about Hausdorff Space:  Definitions, Equivalences, Examples and Counterexamples, Properties, Preregularity Versus Regularity, Variants, Algebra of Functions, Academic Humour

Other articles related to "hausdorff space, space, hausdorff, spaces, hausdorff spaces":

Completely Hausdorff Space - Relation To Other Separation Axioms
... It follows that every completely Hausdorff space is Urysohn and every Urysohn space is Hausdorff ... One can also show that every regular Hausdorff space is Urysohn and every Tychonoff space (=completely regular Hausdorff space) is completely Hausdorff ...
More About Closed Sets
... of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such ... A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A ... In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets ...
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... Every continuous image of a compact space is compact ... Tychonoff's theorem the (arbitrary) product of compact spaces is compact ... A compact subset of a Hausdorff space is closed ...
Weak Hausdorff Space
... In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space ... In particular, every Hausdorff space is weak Hausdorff ... of working with the category of Hausdorff spaces ...
Hausdorff Space - Academic Humour
... Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by open sets ... In the Mathematics Institute of at the University of Bonn, in which Felix Hausdorff researched and lectured, there is a certain room designated the Hausdorff-Raum (Raum stands for both space ...

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