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On the Lattice of Primitive Convergence Structures

Published online by Cambridge University Press:  20 November 2018

C. R. Atherton Jr.
C. R. Atherton Jr.*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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Let S be any set and denote by F(S) the collection of all fiters on S. The collection A(S) of all mappings from F(S) to 2s, 2s being ordered by the dual of its usual ordering, may be regarded as a product of complete Boolean algebras and is, therefore, a complete atomic Boolean algebra [4]. A(S) is called the lattice of primitive convergence structures on S. If q ∈ A(S) and , then is said to q-converge to a point x ∈ S if . The collection of all topologies on S may be identified with a subset of A(S); this subset of A(S) will be denoted by T(S). A more specialized class of primitive convergence structures, and one which properly contains T(S), is C(S), the subcomplete lattice of all convergence structures on S.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 392 - 397
Copyright
Copyright © Canadian Mathematical Society 1971

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