View source for Tikhonov space

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A [[topological space]] satisfying the [[separation axiom]] that every finite set is closed and such that for every closed set $P$ and any point $x$ not in $P$ there exists a continuous real-valued function $f$ on the whole space taking the value 0 at $x$ and the value 1 at every point of $P$. The class of Tikhonov spaces coincides with the class of completely-regular $T_1$-spaces (cf. [[Completely-regular space]]). In a Tikhonov space any two distinct points can be separated by disjoint neighbourhoods (in other words, the [[Hausdorff axiom]] is satisfied), but not every Tikhonov space is normal (cf. [[Normal space]]). A.N. Tikhonov (1929) characterized Tikhonov spaces as subspaces of compact Hausdorff spaces.

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian) {{ZBL|0070.04704}}</TD></TR>
<TR><TD valign="top">[2]</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>
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