# Tietze extension theorem

The**Tietze extension theorem**in topology states that, if

*X*is a normal topological space and

*f*:*A*→**R**

*A*of

*X*into the real numbers carrying the standard topology, then there exists a continuous map

*F*:*X*→**R**

*F*(

*a*) =

*f*(

*a*) for all

*a*in

*A*.

*F*is called a

*continuous extension*of

*f*.

The theorem generalizes Urysohn's lemma and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal.