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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

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