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 $$[\ \ ]_p$$


 * $$[\ \ ]_p:\mathcal{P}(X) \to \mathcal{P}(X)$$

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 its complement $$A = X \setminus U$$ is closed. The collection of all open sets generated by the preclosure operator is a topology; however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.

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 $$[\ \ ]_\text{seq}$$ 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 $$\mathcal{T}_\text{seq}$$ generated by $$[\ \ ]_\text{seq}$$ is equal to $$\mathcal{T},$$ that is, if $$\mathcal{T}_\text{seq} = \mathcal{T}.$$