Urysohn's lemma

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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 generalized by (and usually used in the proof of) the Tietze extension theorem.

The lemma is named after the mathematician Pavel Samuilovich Urysohn.

Discussion[edit]

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 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., we do not, and in general cannot, require that and for outside of and The spaces in which this property holds are the perfectly normal spaces.

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[edit]

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

Sketch of proof[edit]

Illustration of Urysohn's "onion" function.

The procedure is an entirely straightforward application of the definition of normality (once one draws some figures representing the first few steps in the induction described below to see what is going on), beginning with two disjoint closed sets. The clever part of the proof is the indexing of the open sets thus constructed by dyadic fractions.

For every dyadic fraction we are going to construct an open subset of such that:

  1. contains and is disjoint from for all
  2. For the closure of is contained in

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

In order to construct the sets we actually do a little bit more: we construct sets and such that

  • and for all
  • and are open and disjoint for all
  • For is contained in the complement of and the complement of is contained in

Since the complement of is closed and contains the latter condition then implies condition (2) from above.

This construction proceeds by mathematical induction. First define and Since is normal, we can find two disjoint open sets and which contain and respectively. Now assume that and the sets and have already been constructed for Since is normal, for any we can find two disjoint open sets which contain and respectively. Call these two open sets and and verify the above three conditions.

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

See also[edit]

Notes[edit]

  1. ^ Willard 1970 Section 15.

References[edit]

  • Willard, Stephen (2004) [1970]. General Topology (First ed.). 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.

External links[edit]