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.
A preclosure operator on a set
is the power set of
The preclosure operator has to satisfy the following properties: The last axiom implies the following: A set
is closed (with respect to the preclosure) if
is open (with respect to the preclosure) if its complement
The collection of all open sets generated by the preclosure operator is a topology;[1] however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.
The sequential closure operator
seq
{\displaystyle [\ \ ]_{\text{seq}}}
is a preclosure operator.
with respect to which the sequential closure operator is defined, the topological space
is a sequential space if and only if the topology