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''$T_0$-axiom''
  
The weakest of all separation axioms (cf. [[Separation axiom|Separation axiom]]) in general topology; introduced by A.N. Kolmogorov. A topological space satisfies this axiom, i.e. it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k0556704.png" />-space or a Kolmogorov space, if for any two distinct points of the space there exists an open set containing one of the points but not containing the other. If it is required that each of the two (arbitrarily given) points be contained in an open set not containing the other, then one obtains the next stronger separation axiom, called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k0556706.png" />-axiom; topological spaces satisfying it are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k0556708.png" />-spaces. The simplest example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k0556709.png" />-space that is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567010.png" />-space is the connected [[Digon|digon]].
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The weakest of all [[separation axiom]]s in general topology; introduced by A.N. Kolmogorov. A topological space satisfies this axiom, i.e. it is a $T_0$-space or a Kolmogorov space, if for any two distinct points of the space there exists an open set containing one of the points but not containing the other. If it is required that each of the two (arbitrarily given) points be contained in an open set not containing the other, then one obtains the next stronger separation axiom, called the $T_1$-axiom; topological spaces satisfying it are called $T_1$-spaces. The simplest example of a $T_0$-space that is not a $T_1$-space is the connected [[digon]].
  
In a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567011.png" />-space, a singleton need not be closed; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567012.png" />-spaces can be defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567013.png" />-spaces for which all singletons are closed. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567014.png" />-space in which the intersection of any number of open sets is open is called a discrete space (in the broad sense). It is precisely in these spaces that the closure of the union of any family of sets is the same as the union of the closures of these sets. In any discrete space and even in any Kolmogorov space one can define a (partial) order between its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567016.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567018.png" /> is contained in the closure of the singleton consisting of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567019.png" />. Conversely, if one defines in an arbitrary partially ordered set the closure of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567020.png" /> as the set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567021.png" /> and takes for the closure of a set the union of the closures of all of its points, then one obtains a discrete space. Thus, the study of discrete spaces is equivalent to that of partially ordered sets. Simplicial (and more general) complexes in combinatorial topology are important examples of discrete spaces: For two simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567023.png" /> the order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567024.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567025.png" /> is a face (possibly improper) of the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567026.png" />.
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In a $T_0$-space, a [[singleton]] need not be closed; $T_1$-spaces can be defined as $T_0$-spaces for which all singletons are closed. A $T_0$-space in which the intersection of any number of open sets is open is called a discrete space (in the broad sense). It is precisely in these spaces that the closure of the union of any family of sets is the same as the union of the closures of these sets. In any discrete space and even in any Kolmogorov space one can define a (partial) order between its points $x$ and $y$: $x \sqsubseteq y$ if $x$ is contained in the closure of the singleton consisting of the point $y$. Conversely, if one defines in an arbitrary partially ordered set the closure of any point $x$ as the set of all points $x' \le x$ and takes for the closure of a set the union of the closures of all of its points, then one obtains a discrete space. Thus, the study of discrete spaces is equivalent to that of partially ordered sets. Simplicial (and more general) complexes in combinatorial topology are important examples of discrete spaces: For two simplices $x$, $y$ the order relation $x \le y$ means that $x$ is a face (possibly improper) of the simplex $y$.
  
 
====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====
"Natural"  spaces which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567027.png" /> but, as a rule, not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567028.png" /> are the affine schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055670/k05567030.png" /> a commutative ring with identity; cf. [[Affine scheme|Affine scheme]].
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"Natural"  spaces which are $T_0$ but, as a rule, not $T_1$ are the affine schemes $\mathrm{Spec}(A)$, for $A$ a commutative ring with identity; cf. [[Affine scheme]].
  
 
"Discrete spaces in the broad sense" , mentioned above, are commonly called Aleksandrov-discrete spaces; they were first studied in [[#References|[a1]]].
 
"Discrete spaces in the broad sense" , mentioned above, are commonly called Aleksandrov-discrete spaces; they were first studied in [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Alexandrov,  "Diskrete Räume"  ''Mat. Sb.'' , '''1 (43)'''  (1937)  pp. 501–519</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Alexandrov,  "Diskrete Räume"  ''Mat. Sb.'' , '''1 (43)'''  (1937)  pp. 501–519</TD></TR>
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</table>
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Revision as of 17:42, 1 January 2016

$T_0$-axiom

The weakest of all separation axioms in general topology; introduced by A.N. Kolmogorov. A topological space satisfies this axiom, i.e. it is a $T_0$-space or a Kolmogorov space, if for any two distinct points of the space there exists an open set containing one of the points but not containing the other. If it is required that each of the two (arbitrarily given) points be contained in an open set not containing the other, then one obtains the next stronger separation axiom, called the $T_1$-axiom; topological spaces satisfying it are called $T_1$-spaces. The simplest example of a $T_0$-space that is not a $T_1$-space is the connected digon.

In a $T_0$-space, a singleton need not be closed; $T_1$-spaces can be defined as $T_0$-spaces for which all singletons are closed. A $T_0$-space in which the intersection of any number of open sets is open is called a discrete space (in the broad sense). It is precisely in these spaces that the closure of the union of any family of sets is the same as the union of the closures of these sets. In any discrete space and even in any Kolmogorov space one can define a (partial) order between its points $x$ and $y$: $x \sqsubseteq y$ if $x$ is contained in the closure of the singleton consisting of the point $y$. Conversely, if one defines in an arbitrary partially ordered set the closure of any point $x$ as the set of all points $x' \le x$ and takes for the closure of a set the union of the closures of all of its points, then one obtains a discrete space. Thus, the study of discrete spaces is equivalent to that of partially ordered sets. Simplicial (and more general) complexes in combinatorial topology are important examples of discrete spaces: For two simplices $x$, $y$ the order relation $x \le y$ means that $x$ is a face (possibly improper) of the simplex $y$.

References

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


Comments

"Natural" spaces which are $T_0$ but, as a rule, not $T_1$ are the affine schemes $\mathrm{Spec}(A)$, for $A$ a commutative ring with identity; cf. Affine scheme.

"Discrete spaces in the broad sense" , mentioned above, are commonly called Aleksandrov-discrete spaces; they were first studied in [a1].

References

[a1] P.S. [P.S. Aleksandrov] Alexandrov, "Diskrete Räume" Mat. Sb. , 1 (43) (1937) pp. 501–519
How to Cite This Entry:
Kolmogorov axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov_axiom&oldid=13764
This article was adapted from an original article by V.I. Zaitsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article