A set is closed (with respect to the preclosure) if . A set is open (with respect to the preclosure) if its
complement is closed. 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.[2]
The
sequential closure operator is a preclosure operator. Given a topology with respect to which the sequential closure operator is defined, the
topological space is a
sequential space if and only if the topology generated by is equal to that is, if
^ Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of
Sciences, 1966, Theorem 14 A.9
[1].
^S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology,
AMS, Contemporary Mathematics, 2009.
A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin.
ISBN3-540-18178-4.