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A [[Topological space|topological space]] in which every finite set is closed and such that for every closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092850/t0928501.png" /> and any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092850/t0928502.png" /> not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092850/t0928503.png" /> there exists a continuous real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092850/t0928504.png" /> on the whole space taking the value 0 at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092850/t0928505.png" /> and the value 1 at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092850/t0928506.png" />. The class of Tikhonov spaces coincides with the class of completely-regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092850/t0928507.png" />-spaces (cf. [[Completely-regular space|Completely-regular space]]). In a Tikhonov space any two distinct points can be separated by disjoint neighbourhoods (in other words, the Hausdorff separation axiom is satisfied), but not every Tikhonov space is normal (cf. [[Normal space|Normal space]]). A.N. Tikhonov (1929) characterized Tikhonov spaces as subspaces of compact Hausdorff spaces.
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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====
 
====References====
<table><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)</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></table>
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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)</TD></TR>
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<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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Latest revision as of 11:58, 22 January 2021

2010 Mathematics Subject Classification: Primary: 54D15 [MSN][ZBL]

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

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
How to Cite This Entry:
Tikhonov space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tikhonov_space&oldid=15995
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