Tietze Extension Theorem

The Tietze Extension Theorem deals with the problem of extending a continuous real-valued function that is defined on a subspace of a topological space $X$ to a continuous function defined on all of $X$.

Definition

Let $X$ be a normal space and let $A$ be a closed subspace of $X$.

1. Any continuous map of $A$ into the closed interval $[a,b]$ of $\mathbb{R}$ may be extended to a continuous map of all of $X$ onto $[a,b]$.
2. Any continuous map of $A$ into $\mathbb{R}$ may be extended to a continuous map of all of $X$ into $\mathbb{R}$.

Further Reading

• Urysohn Lemma

See Also

• Urysohn Metrization Theorem