Definition:Urysohn Function
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Definition
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.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Completely Regular Spaces