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The Nachibin quasi-uniformity of a bi-Stonian space

Published online by Cambridge University Press:  09 April 2009

P. Fletcher
Affiliation:
Mathematics Department Virginia Polytechnic Institute Blacksburg VA 24061 USA e-mail: Pflectche@math.vt.edu
J. Frith
Affiliation:
Mathematics Department University of Cape Town Rondebosch 7700 South Africa e-mail: jfrith@maths.uct.ac.za
W. Hunsaker
Affiliation:
Mathematics Department Southern Illinois University Carbondale IL 62901 USA e-mail: hunsaker@math.siu.edu
A. Schauerte
Affiliation:
Mathematics Department University of Cape Town Rondebosch 7700 South Africa e-mail: schauert@math.uct.ac.za
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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1996

References

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