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The Nachibin quasi-uniformity of a bi-Stonian space

Part of: Distributive lattices Spaces with richer structures Algebraic logic Special properties Boolean algebras (Boolean rings)

Published online by Cambridge University Press:  09 April 2009

P. Fletcher ,
J. Frith ,
W. Hunsaker  and
A. Schauerte
P. Fletcher
Affiliation:
Mathematics Department Virginia Polytechnic Institute Blacksburg VA 24061 USA e-mail: [email protected]
J. Frith
Affiliation:
Mathematics Department University of Cape Town Rondebosch 7700 South Africa e-mail: [email protected]
W. Hunsaker
Affiliation:
Mathematics Department Southern Illinois University Carbondale IL 62901 USA e-mail: [email protected]
A. Schauerte
Affiliation:
Mathematics Department University of Cape Town Rondebosch 7700 South Africa e-mail: [email protected]
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Abstract

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It is known that every frame is isomorphic to the generalized Gleason algebra of an essentially unique bi-Stonian space (X, σ, τ) in which σ is T0. Let (X, σ, τ) be as above. The specialization order ≤σ, of (X, σ) is τ × τ-closed. By Nachbin's Theorem there is exactly one quasi-uniformity U on X such that ∩U = ≤σ and J(U*) = τ. This quasi-uniformity is compatible with σ and is coarser than the Pervin quasi-uniformity U of (X, σ). Consequently, τ is coarser than the Skula topology of σ and coincides with the Skula topology if and only if U = P.

MSC classification

Secondary: 54E15: Uniform structures and generalizations 54E05: Proximity structures and generalizations 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 06D20: Heyting algebras 03G10: Lattices and related structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 3 , December 1996 , pp. 322 - 326
Copyright
Copyright © Australian Mathematical Society 1996

References

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