The structure of the poset of regular topologies on a set

Ofelia T. Alas, Richard G. Wilson

Abstract

We study the subposet E3(X) of the lattice L1(X) of all T1-topologies on a set X, being the collections of all T3 topologies on X, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. We show that each regular topology which is not R-closed does have such a predecessor and as a corollary we obtain a result of Costantini that each non-compact Tychonoff space has an immediate predecessor in E3. We also consider the problem of when an R-closed topology is maximal R-closed.


Keywords

Lattice of T1-topologies; Poset of T3-topologies; Upper topology; Lower topology; R-closed space; R-minimal space; Submaximal space; Maximal R-closed space; Dispersed space

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References

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