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| In [[topology]], a '''preclosure operator''', or '''Čech closure operator''' is a map between subsets of a set, similar to a topological [[closure operator]], except that it is not required to be [[idempotent]]. That is, a preclosure operator obeys only three of the four [[Kuratowski closure axioms]].
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| == Definition ==
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| A preclosure operator on a set <math>X</math> is a map <math>[\quad]_p</math>
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| :<math>[\quad]_p:\mathcal{P}(X) \to \mathcal{P}(X)</math>
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| where <math>\mathcal{P}(X)</math> is the [[power set]] of <math>X</math>.
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| The preclosure operator has to satisfy the following properties: | |
| # <math> [\varnothing]_p = \varnothing \! </math> (Preservation of nullary unions);
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| # <math> A \subseteq [A]_p </math> (Extensivity);
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| # <math> [A \cup B]_p = [A]_p \cup [B]_p</math> (Preservation of binary unions).
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| The last axiom implies the following:
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| : 4. <math>A \subseteq B</math> implies <math>[A]_p \subseteq [B]_p</math>.
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| ==Topology==
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| A set <math>A</math> is closed (with respect to the preclosure) if <math>[A]_p=A</math>. A set <math>U\subset X</math> is open (with respect to the preclosure) if <math>A=X\setminus U</math> is closed. The collection of all open sets generated by the preclosure operator is a [[topological space|topology]].
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| The [[closure operator]] cl on this topological space satisfies <math>[A]_p\subseteq \operatorname{cl}(A)</math> for all <math>A\subset X</math>.
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| ==Examples==
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| ===Premetrics===
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| Given <math>d</math> a [[premetric]] on <math>X</math>, then
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| :<math>[A]_p=\{x\in X : d(x,A)=0\}</math>
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| is a preclosure on <math>X</math>. | |
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| ===Sequential spaces===
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| The [[sequential closure operator]] <math>[\quad]_\mbox{seq}</math> is a preclosure operator. Given a topology <math>\mathcal{T}</math> with respect to which the sequential closure operator is defined, the topological space <math>(X,\mathcal{T})</math> is a [[sequential space]] if and only if the topology <math>\mathcal{T}_\mbox{seq}</math> generated by <math>[\quad]_\mbox{seq}</math> is equal to <math>\mathcal{T}</math>, that is, if <math>\mathcal{T}_\mbox{seq}=\mathcal{T}</math>.
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| ==See also==
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| * [[Eduard Čech]]
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| ==References==
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| * A.V. Arkhangelskii, L.S.Pontryagin, ''General Topology I'', (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.
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| * B. Banascheski, [http://www.emis.de/journals/CMUC/pdf/cmuc9202/banas.pdf ''Bourbaki's Fixpoint Lemma reconsidered''], Comment. Math. Univ. Carolinae 33 (1992), 303-309.
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| [[Category:Closure operators]]
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