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Difference between revisions of "Functional separability"

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(synonym: completely separated)
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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 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.
 
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====
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" (2nd ed), Springer  (1975) {{ZBL|0306.54002}}</TD></TR>
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</table>

Revision as of 20:28, 14 December 2017


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=42528
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