Urysohn's lemma

The lemma of Urysohn (also called Urysohnsches Lemma ) is a fundamental theorem from the mathematical branch of general topology.

The lemma is named after Pavel Urysohn and published by this 1925. It is often used to construct a continuous function with certain properties. Its wide application possibility based on the fact that many of the most important topological spaces as metric spaces and compact Hausdorff spaces have the vorausgestzte in the lemma normality property.

A generalization is the extension theorem of Tietze dar. In the proof of the Urysohn lemma comes in a decisive manner to bear.

Formulation of the lemma

The lemma states the following:

Comments

1) The lemma of Urysohn says nothing about the values ​​from outside the closed subsets and, but only that and apply. In the event that for disjoint closed and steady always be found with and is called a perfectly normal space.

2) For metric spaces is a continuous function of the above type to specify immediately. To this end, we define two given disjoint closed subsets and the function as follows:

This is the distance from to so

The function is continuous - even uniformly continuous - and the following applies:

Metric spaces are therefore always perfectly normal.

Core statement of the lemma

The core of the lemma of Urysohn lies in the following statement: