Which topologies can have immediate successors in the lattice of T1-topologies?

Authors

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

DOI:

https://doi.org/10.4995/agt.2004.1972

Keywords:

Lattice of T1-topologies, Maximal point, KC-space, Cover of a topology, Upper topology, Lower topology

Abstract

We give a new characterization of those topologies which have an immediate successor or cover in the lattice of T1-topologies on a set and show that certain classes of compact and countably compact topologies do not have covers.

Downloads

Download data is not yet available.

Author Biographies

Ofelia T. Alas, Universidade de Sao Paulo

Instituto de Matemática e Estatística

Richard G. Wilson, Universidad Autónoma Metropolitana

Departamento de Matemáticas

References

O. T. Alas, V. V. Tkachuk and R. G. Wilson, Closures of discrete sets often reflect global properties, Topology Proceedings 25 (2000), 27-44.

A. Bella, Few remarks on spaces which are generated by discrete sets, to appear.

A. Bella and V. I. Malykhin, F-points in countably compact spaces, Applied General Topology 2 no. 1 (2001), 33-37.

E. van Douwen, Applications of maximal topologies, Topology and Its Applications 51 (1993), 125-139. http://dx.doi.org/10.1016/0166-8641(93)90145-4

A. Dow, M. G. Tkachenko, V. V. Tkachuk and R. G. Wilson, Topologies generated by discrete subspaces, Glasnik Mat. Ser. III, 37 (57) no. 1 (2002), 187-210.

R. Engelking, General Topology, (Heldermann Verlag, Berlin, 1989).

L. Gillman and M. Jerison, Rings of Continuous Functions, (Van Nostrand, Princeton, NJ, 1960). http://dx.doi.org/10.1007/978-1-4615-7819-2

I. Juhász, Cardinal Functions in Topology - Ten Years Later, Mathematical Centre Tracts 123, (Amsterdam, 1980).

R. E. Larson and S. Andima, The lattice of topologies; A survey, Rocky Mountain J. of Math. 5 no. 2 (1975), 177-198. http://dx.doi.org/10.1216/RMJ-1975-5-2-177

R. E. Larson and W. J. Thron, Covering relations in the lattice of T1-topologies, Transactions of the American Mathematical Society 168 (1972), 101-111.

L. A. Steen and J. A. Seebach, Counterexamples in Topology, (Springer Verlag, New York, 1978). http://dx.doi.org/10.1007/978-1-4612-6290-9

A. K. Steiner, Complementation in the lattice of T1-topologies, Proceedings of the American Mathematical Society 17 (1966), 884-885.

E. F. Steiner and A. K. Steiner, Topologies with T1-complements, Fundamenta Mathematicae 61 (1967), 23-28.

R. M. Stephenson, Initially -compact and related spaces, in Handbook of Set-theoretic Topology, K. Kunen and J. E. Vaughan, (eds.), (North Holland, Amsterdam, 1984).

M. G. Tkachenko, V. V. Tkachuk, R. G. Wilson and I. V. Yaschenko, No submaximal topology on a countable set is T1-complementary, Proceedings of the American Mathematical Society 128 no. 1 (1999), 287-297. http://dx.doi.org/10.1090/S0002-9939-99-04984-9

R. Valent and R. E. Larson, Basic intervals in the lattice of topologies, Duke Math. J. 39 (1972), 401-411. http://dx.doi.org/10.1215/S0012-7094-72-03948-8

R. C. Walker, The Stone-Cech Compactification, (Springer-Verlag, New York, 1974).

S. Willard, General Topology, (Addison Wesley, Reading, Mass., 1970).

Downloads

How to Cite

[1]
O. T. Alas and R. G. Wilson, “Which topologies can have immediate successors in the lattice of T1-topologies?”, Appl. Gen. Topol., vol. 5, no. 2, pp. 231–242, Oct. 2004.

Issue

Section

Articles