The structure of the poset of regular topologies on a set

Ofelia T. Alas; Richard G. Wilson

doi:10.4995/agt.2011.1695

Authors

Ofelia T. Alas Universidade de Sao Paulo
Richard G. Wilson Universidad Autónoma Metropolitana

DOI:

https://doi.org/10.4995/agt.2011.1695

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

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.

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Author Biographies

Ofelia T. Alas, Universidade de Sao Paulo

Instituto de Matemática e Estatística, Universidade de Sao Paulo, Caixa Postal66281, 05311-970 Sao Paulo, Brasil.

Richard G. Wilson, Universidad Autónoma Metropolitana

Departamento de Matemáticas

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