Definition:Urysohn Function

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Let $T = \struct {S, \tau}$ be a topological space.

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

Let $f: S \to \closedint 0 1$ be a continuous mapping where $\closedint 0 1$ 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:

$\forall a \in A: \map f a = 0$
$\forall b \in B: \map f b = 1$

Also see

  • Results about Urysohn functions can be found here.

Source of Name

This entry was named for Pavel Samuilovich Urysohn.