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The Lattice of all Topologies is Complemented

Published online by Cambridge University Press:  20 November 2018

A. C. M. Van Rooij
A. C. M. Van Rooij*
Affiliation:
Katholieke Universiteit, Nijmegen, Netherlands
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In (3), J. Hartmanis raised the question whether the lattice of all topologies in a given set is complemented and gave the affirmative answer for the case of a finite set. H. Gaifman (2), has extended this result to denumerable sets. Using Gaifman's paper, Anne K. Steiner (4) has proved that the lattice is always complemented. Our aim in this article is to give an alternative proof, independent of Gaifman's results. So far, Steiner's proof has not been available to the author.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 805 - 807
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloq. Publ., Vol. 25 (Amer. Math. Soc, Providence, R.I., 1960).Google Scholar
2. Gaifman, H., The lattice of all topologies, Can. J. Math. 18 (1966), 83–88.Google Scholar
3. Hartmanis, J., On the lattice of topologies, Can. J. Math. 10 (1958), 547–553.Google Scholar
4. Steiner, Anne K., The topological complementation problem, Technical report No. 84, 1965, Univ. of New Mexico.Google Scholar