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.[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.

DiscussionEdit

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 StatementEdit

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 proofEdit

 
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 alsoEdit

NotesEdit

  1. ^ Willard 1970 Section 15.

ReferencesEdit

  • Willard, Stephen (2004) [1970]. General Topology. Dover Books on Mathematics (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 linksEdit