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Remarks on Complementation in the Lattice of all Topologies

Published online by Cambridge University Press:  20 November 2018

Haim Gaifman
Haim Gaifman*
Affiliation:
Hebrew University of Jerusalem
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Our aim is to prove that certain topologies have complements in the lattice of all the topologies on a given set. Lattices of topologies were studied in (1-8). In (7) Hartmanis points out that the lattice of all the topologies on a finite set is complemented and poses the question whether this is so if the set is infinite. A positive answer is given here for denumerable sets. This result was announced in (6). The case of higher powers remains unsettled, although quite a few topologies turn out to have complements. As far as the author knows, no one has proved the existence of a topology that has no complement.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 83 - 88
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bagley, R. W., On the characterization of the lattice of topologies, J. London Math. Soc, 30 (1955), 247–249.Google Scholar
2. Bagley, R. W. and Ellis, David, On the topolattice and permutation group of an infinite set, Math. Japon., 3 (1954), 63–70.Google Scholar
3. Ellis, David, On the topolattice and permutation group of an infinite set, II, Proc. Cambridge Philos. Soc, 50 (1954), 485–487.Google Scholar
4. Balachandran, V. K., On the lattice of convergence topologies, J. Madras Univ. Sect. B, 28 (1958), 129–146.Google Scholar
5. Birkhoff, G., On the combination of topologies, Fund. Math., 26 (1936), 156–166.Google Scholar
6. Gaifman, Haim, The lattice of all topologies en a denumerable set (Abstract), Amer. Math. Soc. Not., 8 (1961), 356.Google Scholar
7. Hartmanis, Juris, On the lattice of topologies, Can. J. Math., 10 (1958), 547–553.Google Scholar
8. Vaidyanathaswami, R., Treatise on set topology (Madras, 1947).Google Scholar