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preclosure

Preclosure operator

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

[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:

  1. [varnothing]_p = varnothing ! (Preservation of nullary unions);
  2. A subseteq [A]_p (Extensivity);
  3. [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.

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.

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

Examples

Premetrics

Given d a prametric on X, then

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

is a preclosure on X.

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

  • 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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