Preclosure operator

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) o 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.

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.

equential 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}.

ee also

* Eduard Čech

References

* A.V. Arkhangelskii, L.S.Pontryagin, "General Topology I", (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.
* 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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