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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: garyr@pegasus.cc.ucf.edu
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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

References

[1]Arens, R. F., ‘A topology for spaces of transformations’, Ann. of Math. 47 (1946), 480–495.CrossRefGoogle Scholar
[2]Brock, P. and Kent, D., ‘Approach spaces, limit tower spaces and probabilistic convergence spaces’, Appl. Categorical Structures 5 (1997), 99–110.CrossRefGoogle Scholar
[3]Brock, P. and Kent, D., ‘Probabilistic convergence spaces and regularity’, Internat. J. Math. Math. Sci. 20 (1997), 637–646.CrossRefGoogle Scholar
[4]Chung, K. L., A course in probability theory (Academic Press, New York, 1974).Google Scholar
[5]Cook, C. and Fisher, H., ‘On equicontinuity and continuous convergence’, Math. Ann. 159 (1965), 94–104.CrossRefGoogle Scholar
[6]Cook, C. and Fisher, H., ‘Regular convergence spaces’, Math. Ann. 174 (1967), 1–7.CrossRefGoogle Scholar
[7]Floresque, L., ‘Probabilistic convergence structures’, Aequationes Math. 38 (1989), 123–145.CrossRefGoogle Scholar
[8]Kent, D., McKennon, K., Richardson, G. and Schroder, M., ‘Continuous convergence in C(X)’, Pacific J. Math. 52 (1974), 271–279.CrossRefGoogle Scholar
[9]Lowen, E. and Lowen, R., ‘A quasi topos containing CONV and MET as full subcategories’, Internat. J. Math. Math Sci. 11 (1988), 417–438.CrossRefGoogle Scholar
[10]Lowen, R., ‘Approach spaces: A common subcategory of TOP and MET’, Math. Nachr. 141 (1989), 183–226.CrossRefGoogle Scholar
[11]Menger, K., ‘Statistical metrics’, Proc. Nat. Acad. Sci. 28 (1942), 535–537.CrossRefGoogle ScholarPubMed
[12]Richardson, G. and Kent, D., ‘The regularity series of a convergence space’, Bull. Austral. Math. Soc. 13 (1975), 21–44.CrossRefGoogle Scholar
[13]Richardson, G. and Kent, D., ‘Probabilistic convergent spaces’, J. Austral. Math. Soc. (Series A) 61 (1996), 1–21.Google Scholar
[14]Schweizer, B. and Sklar, A., Probabilistic metric spaces (North-Holland, New York, 1983).Google Scholar