# Urysohn Lemma

The **Urysohn Lemma** asserts the existence of certain real-valued continuous functions on a normal space $X$. The Urysohn Lemma has several significant implications, among them the Tietze Extension Theorem and the Urysohn Metrization Theorem.

## Formal Statement

Let $X$ be a normal space and let $A$ and $B$ be disjoint closed subsets of $X$. Let $[a,b]$ be a closed interval in the real line. Then there exists a continuous map $$f : X \to [a,b]$$ such that $f(x) = a$ for every $x$ in $A$, and $f(x) = b$ for every $x$ in $B$.^{[1]}

## Further Reading

## See Also

## References

- ↑ Munkres, James R. Topology, 2015. pg. 205.