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T-regular probabilistic convergence spaces

Part of: Generalities Spaces with richer structures

Published online by Cambridge University Press:  09 April 2009

J. Minkler ,
G. Minkler  and
G. Richardson
G. Richardson
Affiliation:
Department of Mathematics University of Central FloridaOrlando, FL 32816-1364USA e-mail address: [email protected]
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Abstract

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A probabilistic convergence structure assigns a probability that a given filter converges to a given element of the space. The role of the t-norm (triangle norm) in the study of regularity of probabilistic convergence spaces is investigated. Given a probabilistic convergence space, there exists a finest T-regular space which is coarser than the given space, and is referred to as the ‘T-regular modification’. Moreover, for each probabilistic convergence space, there is a sequence of spaces, indexed by nonnegative ordinals, whose first term is the given space and whose last term is its T-regular modification. The T-regular modification is illustrated in the example involving ‘convergence with probability λ’ for several t-norms. Suitable function space structures in terms of a given t-norm are also considered.

Keywords

Convergence spaceprobabilistic convergence spaceT-regular space.

MSC classification

Secondary: 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.) 54E70: Probabilistic metric spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 2 , April 1998 , pp. 210 - 221
Copyright
Copyright © Australian Mathematical Society 1998

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