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Functional separability

From Encyclopedia of Mathematics
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The property of two sets and in a topological space requiring the existence of a continuous real-valued function on such that the closures of the sets and (relative to the usual topology on the real line ) 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.

References

[1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[2] J.L. Kelley, "General topology" , Springer (1975)
How to Cite This Entry:
Functional separability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_separability&oldid=33205
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