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 U\subset X is open (with respect to the preclosure) if A=X\setminus 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]_p\subseteq \operatorname{cl}(A) for all A\subset X.

Examples

Premetrics

Given d a premetric on X, then

[A]_p=\{x\in 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

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