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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.
## Definition

A preclosure operator on a set $X$ is a map $[quad]\_p$ ## Topology

A set $A$ is closed (with respect to the preclosure) if $[A]\_p=A$. A set $Usubset\; X$ is open (with respect to the preclosure) if $A=Xsetminus\; U$ is closed. The collection of all open sets generated by the preclosure operator is a topology.## Examples

### Premetrics

Given $d$ a prametric on $X$, then ### Sequential spaces

The sequential closure operator $[quad]\_mbox\{seq\}$ is a preclosure operator. Given a topology $mathcal\{T\}$ with respect to which the sequential closure operator is defined, the topological space $(X,mathcal\{T\})$ is a sequential space if and only if the topology $mathcal\{T\}\_mbox\{seq\}$ generated by $[quad]\_mbox\{seq\}$ is equal to $mathcal\{T\}$, that is, if $mathcal\{T\}\_mbox\{seq\}=mathcal\{T\}$.
## See also

## References

- $[quad]\_p:mathcal\{P\}(X)\; to\; mathcal\{P\}(X)$

where $mathcal\{P\}(X)$ is the power set of $X$.

The preclosure operator has to satisfy the following properties:

- $[varnothing]\_p\; =\; varnothing\; !$ (Preservation of nullary unions);
- $A\; subseteq\; [A]\_p$ (Extensivity);
- $[A\; cup\; B]\_p\; =\; [A]\_p\; cup\; [B]\_p$ (Preservation of binary unions).

The last axiom implies the following:

- 4. $A\; subseteq\; B$ implies $[A]\_p\; subseteq\; [B]\_p$.

The closure operator cl on this topological space satisfies $[A]\_psubseteq\; operatorname\{cl\}(A)$ for all $Asubset\; X$.

- $[A]\_p=\{xin\; X\; :\; d(x,A)=0\}$

is a preclosure on $X$.

- A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.
- B. Banascheski, Bourbaki's Fixpoint Lemma reconsidered, Comment. Math. Univ. Carolinae 33 (1992), 303-309.

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Last updated on Saturday June 21, 2008 at 08:21:06 PDT (GMT -0700)

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