Preclosure operator
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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.
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. [\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 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.
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 [\ \ ]_\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}.
See also
Eduard Čech
References
A.V. Arkhangelskii, L.S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. . B. Banaschewski, Bourbaki's Fixpoint Lemma reconsidered, Comment. Math. Univ. Carolinae 33 (1992), 303–309.
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