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A condition imposed on a [[Topological space|topological space]], expressing the requirement that some disjoint (i.e. not having common points) sets can be topologically separated from each other in a specific way. The simplest (i.e. weakest) of these axioms apply only to one-point sets, i.e. to the points of a space. These are the so-called axioms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s0845202.png" /> (Kolmogorov's separation axiom, cf. also [[Kolmogorov space|Kolmogorov space]]; [[Kolmogorov axiom|Kolmogorov axiom]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s0845204.png" />. The next in line are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s0845205.png" /> (Hausdorff's separation axiom), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s0845206.png" /> (regularity axiom) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s0845207.png" /> (normality axiom), which require, respectively, that every two different points (axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s0845208.png" />), every point and every closed set not containing it (axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s0845209.png" />), and every two disjoint closed sets (axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452010.png" />) can be separated by neighbourhoods, i.e. are contained in disjoint open sets of the given space.
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A condition imposed on a [[Topological space|topological space]], expressing the requirement that some disjoint (i.e. not having common points) sets can be topologically separated from each other in a specific way. The simplest (i.e. weakest) of these axioms apply only to one-point sets, i.e. to the points of a space. These are the so-called axioms $T_0$ (Kolmogorov's separation axiom, cf. also [[Kolmogorov space|Kolmogorov space]]; [[Kolmogorov axiom|Kolmogorov axiom]]) and $T_1$. The next in line are $T_2$ (Hausdorff's separation axiom), $T_3$ (regularity axiom) and $T_4$ (normality axiom), which require, respectively, that every two different points (axiom $T_2$), every point and every closed set not containing it (axiom $T_3$), and every two disjoint closed sets (axiom $T_4$) can be separated by neighbourhoods, i.e. are contained in disjoint open sets of the given space.
  
A topological space which satisfies the axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452012.png" />, is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452014.png" />-space; a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452016.png" />-space is also called a Hausdorff space (cf. [[Hausdorff space|Hausdorff space]]), and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452018.png" />-space is called regular (cf. [[Regular space|Regular space]]); a Hausdorff <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452020.png" />-space is always regular and is called normal (cf. [[Normal space|Normal space]]).
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A topological space which satisfies the axiom $T_i$, $i=2,3,4$, is called a $T_i$-space; a $T_2$-space is also called a Hausdorff space (cf. [[Hausdorff space|Hausdorff space]]), and a $T_3$-space is called regular (cf. [[Regular space|Regular space]]); a Hausdorff $T_4$-space is always regular and is called normal (cf. [[Normal space|Normal space]]).
  
Functional separation is of particular significance. Two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452022.png" /> in a given topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452023.png" /> are said to be functionally separated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452024.png" /> if there exists a real-valued bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452025.png" />, defined throughout the space, which takes one value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452026.png" /> at all points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452027.png" />, and a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452028.png" />, different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452029.png" />, at all points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452030.png" />. It can always be supposed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452032.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452033.png" /> at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452034.png" />.
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Functional separation is of particular significance. Two sets $A$ and $B$ in a given topological space $X$ are said to be ''functionally separated'' (or ''completely separated'') in $X$ if there exists a real-valued bounded continuous function $f$, defined throughout the space, which takes one value $a$ at all points of the set $A$, and a value $b$, different from $a$, at all points of the set $B$. It can always be supposed that $a=0$, $b=1$, and that $0\leq f(x)\leq1$ at all points $x\in X$.
  
Two functionally-separable sets are always separable by neighbourhoods, but the converse is not always true. However, Urysohn's lemma holds: In a normal space, every two disjoint closed sets are functionally separable. A space in which every point is functionally separable from every closed set not containing it is called completely regular (cf. [[Completely-regular space|Completely regular space]]). A completely regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452035.png" />-space is called a [[Tikhonov space|Tikhonov space]].
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Two functionally separable sets are always separable by neighbourhoods, but the converse is not always true. However, Urysohn's lemma holds: In a normal space, every two disjoint closed sets are functionally separable. A space in which every point is functionally separable from every closed set not containing it is called completely regular (cf. [[Completely-regular space|Completely regular space]]). A completely regular $T_2$-space is called a [[Tikhonov space]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
The reader is warned that there is not really one convention here. There are authors who equate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452036.png" /> and regularity, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452037.png" /> and normality and take both to include the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452038.png" />-property, e.g., [[#References|[a1]]].
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The reader is warned that there is not really one convention here. There are authors who equate $T_3$ and regularity, and $T_4$ and normality and take both to include the $T_1$-property, e.g., [[#References|[a1]]].
  
 
In [[#References|[a2]]] one finds the convention that  "T3=regular+T1"  and  "T4=normal+T1" , where [[#References|[a3]]] adopts  "regular=T3+T1"  and  "normal=T4+T1" .
 
In [[#References|[a2]]] one finds the convention that  "T3=regular+T1"  and  "T4=normal+T1" , where [[#References|[a3]]] adopts  "regular=T3+T1"  and  "normal=T4+T1" .
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The standpoint of [[#References|[a1]]] seems to be the most widely accepted.
 
The standpoint of [[#References|[a1]]] seems to be the most widely accepted.
  
The adjective  "completely regular"  is often associated with the letter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084520/s08452040.png" />.
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The adjective  "completely regular"  is often associated with the letter $T_{3\frac12}$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. von Querenburg,  "Mengentheoretische Topologie" , Springer  (1973)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  B. von Querenburg,  "Mengentheoretische Topologie" , Springer  (1973)</TD></TR>
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</table>
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====Comments====
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The axiom $T_D$ is intermediate between $T_0$ and $T_1$: it states that every [[singleton]] is a [[locally closed set]]. 
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====References====
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<table>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  Peter T. Johnstone ''Sketches of an elephant'' Oxford University Press (2002)  ISBN 0198534256 {{ZBL|1071.18001}}</TD></TR>
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</table>
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[[Category:General topology]]

Latest revision as of 18:00, 14 December 2017

A condition imposed on a topological space, expressing the requirement that some disjoint (i.e. not having common points) sets can be topologically separated from each other in a specific way. The simplest (i.e. weakest) of these axioms apply only to one-point sets, i.e. to the points of a space. These are the so-called axioms $T_0$ (Kolmogorov's separation axiom, cf. also Kolmogorov space; Kolmogorov axiom) and $T_1$. The next in line are $T_2$ (Hausdorff's separation axiom), $T_3$ (regularity axiom) and $T_4$ (normality axiom), which require, respectively, that every two different points (axiom $T_2$), every point and every closed set not containing it (axiom $T_3$), and every two disjoint closed sets (axiom $T_4$) can be separated by neighbourhoods, i.e. are contained in disjoint open sets of the given space.

A topological space which satisfies the axiom $T_i$, $i=2,3,4$, is called a $T_i$-space; a $T_2$-space is also called a Hausdorff space (cf. Hausdorff space), and a $T_3$-space is called regular (cf. Regular space); a Hausdorff $T_4$-space is always regular and is called normal (cf. Normal space).

Functional separation is of particular significance. Two sets $A$ and $B$ in a given topological space $X$ are said to be functionally separated (or completely separated) in $X$ if there exists a real-valued bounded continuous function $f$, defined throughout the space, which takes one value $a$ at all points of the set $A$, and a value $b$, different from $a$, at all points of the set $B$. It can always be supposed that $a=0$, $b=1$, and that $0\leq f(x)\leq1$ at all points $x\in X$.

Two functionally separable sets are always separable by neighbourhoods, but the converse is not always true. However, Urysohn's lemma holds: In a normal space, every two disjoint closed sets are functionally separable. A space in which every point is functionally separable from every closed set not containing it is called completely regular (cf. Completely regular space). A completely regular $T_2$-space is called a Tikhonov space.

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)


Comments

The reader is warned that there is not really one convention here. There are authors who equate $T_3$ and regularity, and $T_4$ and normality and take both to include the $T_1$-property, e.g., [a1].

In [a2] one finds the convention that "T3=regular+T1" and "T4=normal+T1" , where [a3] adopts "regular=T3+T1" and "normal=T4+T1" .

The standpoint of [a1] seems to be the most widely accepted.

The adjective "completely regular" is often associated with the letter $T_{3\frac12}$.

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
[a2] J.L. Kelley, "General topology" , Springer (1975)
[a3] B. von Querenburg, "Mengentheoretische Topologie" , Springer (1973)

Comments

The axiom $T_D$ is intermediate between $T_0$ and $T_1$: it states that every singleton is a locally closed set.

References

[b1] Peter T. Johnstone Sketches of an elephant Oxford University Press (2002) ISBN 0198534256 Zbl 1071.18001
How to Cite This Entry:
Separation axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separation_axiom&oldid=17073
This article was adapted from an original article by V.I. Zaitsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article