# Tietze extension theorem

The

**Tietze extension theorem** in

topology states that, if

*X* is a

normal topological space and

*f* : *A* `->` **R**

is a

continuous map from a

closed subset *A* of

*X* into the

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

*F* : *X* `->` **R**

with

*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.