# Definition:Urysohn Function

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A, B \subseteq S$ such that $A \cap B = \varnothing$.

Let $f: S \to \left[{0 \,.\,.\, 1}\right]$ be a continuous mapping where $\left[{0\,.\,.\, 1}\right]$ is the closed unit interval.

Then $f$ is an Urysohn function for $A$ and $B$ if and only if:

$f {\restriction_A} = 0, f {\restriction_B} = 1$

that is:

$\forall a \in A: f \left({a}\right) = 0$
$\forall b \in B: f \left({b}\right) = 1$

## Source of Name

This entry was named for Pavel Samuilovich Urysohn.