The structure of the poset of regular topologies on a set
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
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.
O. T. Alas, S. Hern´andez, M. Sanchis, M. G. Tkachenko and R. G. Wilson, Adjacency in the partial orders of Tychonoff, regular and locally compact topologies, Acta Math. Hungar. 112, no. 3 (2006), 2005–2025.
O. T. Alas, M. G. Tkachenko and R. G. Wilson, Which topologies have immediate predecessors in the poset of Hausdorff topologies?, Houston Journal Math., to appear.
O. T. Alas and R. G. Wilson, Which topologies can have immediate successors in the lattice of T1-topologies?, Appl. Gen. Topol. 5, no. 2 (2004), 231–242.
M. Berri, J. Porter and R. M. Stephenson, A survey of minimal topological spaces, Proc. Kanpur Conference, 1968.
N. Carlson, Lower and upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007), 619–624. http://dx.doi.org/10.1016/j.topol.2006.08.003
C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008), 187–225.
R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
L. M. Friedler, M. Girou, D. H. Pettey and J. R. Porter, A survey of R-, U-, and CH-closed spaces, Topology Proc. 17 (1992), 71–96.
S. H. Hechler, Two R-closed spaces revisited, Proc. Amer. Math. Soc. 56 (1976), 303–309.
R. E. Larson and W. J. Thron, Covering relations in the lattice of T1-topologies, Trans. Amer. Math. Soc. 168 (1972), 101–111.
D. W. McIntyre and S. W. Watson, Finite intervals in the partial orders of zero-dimensional, Tychonoff and regular topologies, Topology Appl. 139 (2004), 23–36. http://dx.doi.org/10.1016/j.topol.2003.08.010
J. Porter, R. M. Stephenson and R. G. Woods, Maximal feebly compact spaces, Topology Appl. 52 (1993), 203–219. http://dx.doi.org/10.1016/0166-8641(93)90103-K
J. Porter and R. G. Woods, Extensions and Absolutes of Topological Spaces, Springer Verlag, New York, 1987.
C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147. http://dx.doi.org/10.1090/S0002-9947-1966-0203679-7
R. M. Stephenson, Two R-closed spaces, Canadian J. Math. 24 (1972), 286–292. http://dx.doi.org/10.4153/CJM-1972-023-5
R. Valent and R. E. Larson, Basic intervals in the lattice of topologies, Duke Math. J. 379 (1972), 401–411. http://dx.doi.org/10.1215/S0012-7094-72-03948-8
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