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Metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space {\displaystyle (X,{\mathcal {T}})} is said to be metrizable if there is a metric d : X × X → [ 0 , ∞ ) {\displaystyle d\colon X\times X\to [0,\infty )} such that the topology induced by d is T {\displaystyle {\mathcal {T}}} .
Pavel Urysohn
Pavel Samuilovich Urysohn (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are fundamental results in topology. His name is also commemorated in the terms Urysohn universal space, Fréchet–Urysohn space, Menger–Urysohn dimension and Urysohn integral equation.
Tietze extension theorem
In topology, the Tietze extension theorem states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
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.
Urysohn and completely Hausdorff spaces
In topology, a discipline within mathematics, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a topological space in which any two distinct points can be separated by a continuous function.
Urysohn universal space
The Urysohn universal space is a certain metric space that contains all separable metric spaces in a particularly nice manner. This mathematics concept is due to Pavel Samuilovich Urysohn.
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