Urysohn Lemma

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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 Tieze 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


  1. Munkres, James R. Topology, 2015. pg. 205.