##  [Urysohn's Lemma](/urysohns-lemma-0) 

 Definition

In a normal topological space X, for any two disjoint closed sets A and B there exists a continuous function f: X → [0,1] such that f equals 0 on A and 1 on B; this constructs a continuous separator between closed sets.

 

 

 

 

 

 





## Principle

Principle

Normality (ability to separate closed sets by disjoint open neighborhoods) permits the construction of continuous maps that interpolate prescribed values on disjoint closed sets, realizing a basic partition-of-unity phenomenon.

 

 

 

 

 





## Demonstration

Demonstration

In a metric space define f(x) = d(x,A)/(d(x,A)+d(x,B)) where d denotes distance; this continuous function takes value 0 on A and 1 on B and demonstrates Urysohn's lemma in the metric setting.

 

 

 

 

## Misapplication

Misapplication

Attempting to construct such an f in a non-normal space can fail; spaces that are not normal (e.g., certain product topologies or pathological examples) may contain disjoint closed sets with no continuous separator.

 

 

 

 

 





## Consequence

Consequence

Urysohn's lemma is a stepping stone to the Tietze extension theorem and to partitions of unity: it provides concrete continuous functions used in embedding and extension arguments in topology and analysis.

 

 

 

 

## Reversal

Reversal

If no continuous function can separate two disjoint closed sets, the space cannot be normal; the failure of Urysohn's conclusion gives a certificate of non-normality.

 

 

 

 

 





## Boundary

Boundary

The lemma requires normality of the space (T1 + every two disjoint closed sets have disjoint neighborhoods); it does not hold in merely Hausdorff or in arbitrary topological spaces without additional separation axioms.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Urysohn's lemma is closely related to but weaker than extension theorems: it produces a separating function with fixed boundary values, whereas Tietze guarantees extension of arbitrary continuous functions from closed subsets to the whole normal space.

 

 

 

 

 





## Synthesis

Synthesis

Urysohn's lemma translates the separation property of normal spaces into the existence of explicit continuous interpolants to [0,1], forming a practical tool to separate and extend functions in topological constructions.