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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102801.png" /> be a set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102802.png" /> the set of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102803.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102804.png" /> is a closure operation if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102805.png" />:
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Let $S$ be a set and $\mathcal{P}S$ the set of subsets of $S$. A function $K : \mathcal{P}S \rightarrow \mathcal{P}S$ is a ''closure operation'' if for all $A,B \in \mathcal{P}S$:
  
K1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102806.png" />;
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K1) $A \subseteq K(A)$;
  
K2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102807.png" />;
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K2) $K(K(A)) \subseteq K(A)$;
  
K3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102808.png" />;
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K3) $K(A) \subseteq K(A\cup B)$;
  
K4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c1102809.png" />;
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K4) $K(A\cup B) \subseteq K(A) \cup K(B)$;
  
K5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028010.png" />. These are the Kuratowski closure axioms, and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028011.png" /> satisfying these axioms is called a Kuratowski closure operator (or Kuratowski closure operation).
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K5) $K(\emptyset) = \emptyset$.  
  
A pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028012.png" /> is a closure space if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028013.png" /> satisfies:
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These are the ''Kuratowski closure axioms'', and a function $K$ satisfying these axioms is called a ''Kuratowski closure operator'' (or Kuratowski closure operation).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028014.png" /></td> </tr></table>
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A pair $(S,\mathrm{Cl})$ is a ''closure space'' if and only if $\mathrm{Cl}$ satisfies:
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$$
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A \subseteq \mathrm{Cl}(B) \Leftrightarrow \mathrm{Cl}(A) \subseteq \mathrm{Cl}(B) \ .
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$$
  
This condition is equivalent to K1)–K3). A closure space that satisfies K5) is a Fréchet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028016.png" />-space. A Fréchet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028017.png" />-space that satisfies K4) is a [[Topological space|topological space]].
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This condition is equivalent to K1)–K3). A closure space that satisfies K5) is a Fréchet $V$-space. A Fréchet $V$-space that satisfies K4) is a [[topological space]].
  
A Čech closure operator is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028018.png" /> such that
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A Čech closure operator is a mapping $C : \mathcal{P}S \rightarrow \mathcal{P}S$ such that
  
C1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028019.png" />;
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C1) $C(\emptyset) = \emptyset$;
  
C2) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028020.png" /> K1)) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028021.png" />;
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C2) (${=}$ K1)) $A \subseteq C(A)$;
  
C3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110280/c11028022.png" />. A set endowed with a Čech closure operator is called a pre-topological space.
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C3) $C(A \cup B) = C(A) \cup C(B)$.  
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A set endowed with a Čech closure operator is called a ''pre-topological space''.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Martin,  S. Pollard,  "Closure spaces and logic" , Kluwer Acad. Publ.  (1996)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand–Reinhold (1955)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Dikranjan,  W. Tholin,  "Categorical structures of closure operators" , Kluwer Acad. Publ.  (1996)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Martin,  S. Pollard,  "Closure spaces and logic" , Kluwer Acad. Publ.  (1996)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Graduate Texts in Mathematics '''27''' Springer (1975) ISBN 0-387-90125-6  {{ZBL|0306.54002}}</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Dikranjan,  W. Tholin,  "Categorical structures of closure operators" , Kluwer Acad. Publ.  (1996)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Revision as of 07:35, 12 December 2016

Let $S$ be a set and $\mathcal{P}S$ the set of subsets of $S$. A function $K : \mathcal{P}S \rightarrow \mathcal{P}S$ is a closure operation if for all $A,B \in \mathcal{P}S$:

K1) $A \subseteq K(A)$;

K2) $K(K(A)) \subseteq K(A)$;

K3) $K(A) \subseteq K(A\cup B)$;

K4) $K(A\cup B) \subseteq K(A) \cup K(B)$;

K5) $K(\emptyset) = \emptyset$.

These are the Kuratowski closure axioms, and a function $K$ satisfying these axioms is called a Kuratowski closure operator (or Kuratowski closure operation).

A pair $(S,\mathrm{Cl})$ is a closure space if and only if $\mathrm{Cl}$ satisfies: $$ A \subseteq \mathrm{Cl}(B) \Leftrightarrow \mathrm{Cl}(A) \subseteq \mathrm{Cl}(B) \ . $$

This condition is equivalent to K1)–K3). A closure space that satisfies K5) is a Fréchet $V$-space. A Fréchet $V$-space that satisfies K4) is a topological space.

A Čech closure operator is a mapping $C : \mathcal{P}S \rightarrow \mathcal{P}S$ such that

C1) $C(\emptyset) = \emptyset$;

C2) (${=}$ K1)) $A \subseteq C(A)$;

C3) $C(A \cup B) = C(A) \cup C(B)$.

A set endowed with a Čech closure operator is called a pre-topological space.

References

[a1] N.M. Martin, S. Pollard, "Closure spaces and logic" , Kluwer Acad. Publ. (1996)
[a2] J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
[a3] D. Dikranjan, W. Tholin, "Categorical structures of closure operators" , Kluwer Acad. Publ. (1996)
How to Cite This Entry:
Closure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closure_space&oldid=39981
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article