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.[1]

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.

Discussion

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Two sets separated by neighborhoods.

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.

Formal statement

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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  

Proof sketch

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Illustration of the first few sets built as part of the proof.

The proof proceeds by repeatedly applying the following alternate characterization of normality. If   is a normal space,   is an open subset of  , and   is closed, then there exists an open   and a closed   such that  .

Let   and   be disjoint closed subsets of  . The main idea of the proof is to repeatedly apply this characterization of normality to   and  , continuing with the new sets built on every step.

The sets we build are indexed by dyadic fractions. For every dyadic fraction  , we construct an open subset   and a closed subset   of   such that:

  •   and   for all  ,
  •   for all  ,
  • For  ,  .

Intuitively, the sets   and   expand outwards in layers from  :

 

This construction proceeds by mathematical induction. For the base step, we define two extra sets   and  .

Now assume that   and that the sets   and   have already been constructed for  . Note that this is vacuously satisfied for  . Since   is normal, for any  , we can find an open set and a closed set such that

 

The above three conditions are then verified.

Once we have these sets, we define   if   for any  ; otherwise   for every  , where   denotes the infimum. Using the fact that the dyadic rationals are dense, it is then not too hard to show that   is continuous and has the property   and   This step requires the   sets in order to work.

The Mizar project has completely formalised and automatically checked a proof of Urysohn's lemma in the URYSOHN3 file.

See also

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Notes

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  1. ^ Willard 1970 Section 15.

References

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  • Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
  • Willard, Stephen (1970). General Topology. Dover Publications. ISBN 0-486-43479-6.
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