View source for Functional separability

{{TEX|done}}{{MSC|54D15}}

''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 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====
<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>