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On the lattice equivalence of topological spaces

Published online by Cambridge University Press:  09 April 2009

P. D. Finch
P. D. Finch
Affiliation:
Monash University
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A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 4 , November 1966 , pp. 495 - 511
Copyright
Copyright © Australian Mathematical Society 1966

References

[1]Birkhoff, C., Lattice theory, Rev. Ed. Amer. Math. Soc. Colloquium Publ. New York 1948.Google Scholar
[2]Büchi, J. Richard, ‘Representations of complete lattices by sets’, Portugaliae Math. 11 (1952), 151–167.Google Scholar
[3]Stelleckii, I. V., ‘On complete lattices represented by sets’, Uspeki Mat. Nauk. (NS) 12, No. 6 (78) (1957), 177–80.Google Scholar
[4]Thron, W. J., ‘Lattice equivalence of topological spaces’, Duke Math. Journ. 29 (1962), 671–679.CrossRefGoogle Scholar
[5]V´clav, V., ‘A remark on complete lattices represented by sets’, Casopis. Pest Mat. 87 (1962), 76–80.Google Scholar