Concept
Hausdorff space
Related concepts (25)
Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary. If is a normal space and is a continuous map from a closed subset of into the real numbers carrying the standard topology, then there exists a of to that is, there exists a map continuous on all of with for all Moreover, may be chosen such that that is, if is bounded then may be chosen to be bounded (with the same bound as ).
History of the separation axioms
The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Before the current general definition of topological space, there were many definitions offered, some of which assumed (what we now think of as) some separation axioms. For example, the definition given by Felix Hausdorff in 1914 is equivalent to the modern definition plus the Hausdorff separation axiom.
Urysohn's lemma
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compact Hausdorff spaces are normal. The lemma is generalised by (and usually used in the proof of) the Tietze extension theorem. The lemma is named after the mathematician Pavel Samuilovich Urysohn.
Convergence space
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as , that do not arise from any topological space.
Cauchy space
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. The of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces. Throughout, is a set, denotes the power set of and all filters are assumed to be proper/non-degenerate (i.