The structure of the poset of regular topologies on a set


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



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


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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How to Cite

O. T. Alas and R. G. Wilson, “The structure of the poset of regular topologies on a set”, Appl. Gen. Topol., vol. 12, no. 1, pp. 1–13, Apr. 2011.



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