Tietze extension theorem

The extension theorem of Tietze ( Tietze English ('s ) extension theorem ), also known as Tietze extension theorem of or as a set of Tietze - Urysohn, is a set from the mathematical branch of topology. He is normal topological spaces with continuous sequels in relationship. Was released the record in 1915 by Heinrich Tietze.

The theorem is a generalization of the Urysohn lemma and can be applied in many cases, since all metric spaces and all compact Hausdorff spaces are normal.

Extension theorem of Tietze

A topological space is then a normal room, if at all on a closed subset of well-defined, continuous function

A continuous function

Exists with, that is, for everyone. The function is called continuous extension of.

This is a pure existence theorem. With few exceptions, such a continuous extension is not unique, ie it can cause the function is more than one function with the desired property type.

Stronger version

The extension theorem of Tietze can be in an even stronger version formulated:

A topological space is then and only then a normal space if for any continuous map of the form with a closed and an always exists from intervals of existing product space a continuous extension.

For the applications of the theorem in particular, the case is significant.

Example

In metric spaces can continue to be stated explicitly be completed and not negative. Then

A continuous extension to the whole of.