Difference between revisions of "Functional separability"
From Encyclopedia of Mathematics
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| − | The property of two sets | + | {{TEX|done}}{{MSC|54D15}} |
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| + | ''complete separability'' | ||
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| + | 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 space|completely regular]] if every closed set is separable from each one-point set that does not intersect it. A space is [[Normal space|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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.L. Kelley, "General topology" , Springer (1975)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) {{ZBL|0568.54001}}</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> J.L. Kelley, "General topology" (2nd ed), Springer (1975) {{ZBL|0306.54002}}</TD></TR> | ||
| + | </table> | ||
Latest revision as of 20:44, 14 December 2017
2020 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.
References
| [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: http://encyclopediaofmath.org/index.php?title=Functional_separability&oldid=14738
Functional separability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_separability&oldid=14738
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