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Concept
Urysohn's lemma
Summary
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.
Two subsets and of a topological space are said to be separated by neighbourhoods if there are neighbourhoods of and of that are disjoint. In particular and are necessarily disjoint.
Two plain subsets and are said to be separated by a continuous function if there exists a continuous function from into the unit interval such that for all and for all Any such function is called a Urysohn function for and In particular and are necessarily disjoint.
It follows that if two subsets and are separated by a function then so are their closures. Also it follows that if two subsets and are separated by a function then and are separated by neighbourhoods.
A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.
The sets and need not be precisely separated by , i.e., it is not necessary and guaranteed that and for outside and A topological space in which every two disjoint closed subsets and are precisely separated by a continuous function is perfectly normal.
Urysohn's lemma has led to the formulation of other topological properties such as the 'Tychonoff property' and 'completely Hausdorff spaces'. For example, a corollary of the lemma is that normal T1 spaces are Tychonoff.
A topological space is normal if and only if, for any two non-empty closed disjoint subsets and of there exists a continuous map such that and
The proof proceeds by repeatedly applying the following alternate characterization of normality.
Official source
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