Functional separability

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2010 Mathematics Subject Classification: Primary: 54D15 [MSN][ZBL]

complete separability

The property of two sets $A$ and $B$ in a topological space $X$ requiring the existence of a continuous real-valued function $f$ on $X$ such that the closures of the sets $f(A)$ and $f(B)$ (relative to the usual topology on the real line $\mathbf R$) do not intersect. For example, a space is completely regular if every closed set is separable from each one-point set that does not intersect it. A space is normal if every two closed non-intersecting subsets of it are functionally separable. If every two (distinct) one-point sets in a space are functionally separable, then the space is called functionally Hausdorff. The content of these definitions is unchanged if, instead of continuous real-valued functions, one takes continuous mappings into the plane, into an interval or into the Hilbert cube.


[1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) Zbl 0568.54001
[2] J.L. Kelley, "General topology" (2nd ed), Springer (1975) Zbl 0306.54002
How to Cite This Entry:
Functional separability. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article