A representation theorem for fuzzy pseudometrics

Authors:
I. Mardones-Pérez;M. A. de Prada Vicente
Affiliations:
Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea UPV/EHU, Apdo. 644, 48080 Bilbao, Spain;Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea UPV/EHU, Apdo. 644, 48080 Bilbao, Spain
Venue:
Fuzzy Sets and Systems
Year:
2012

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Abstract

In this paper, we show that there exists a one to one correspondence between a certain class of fuzzy pseudometrics (in the sense of Kramosil and Michalek) and [0,1)-indexed families of ordinary pseudometrics satisfying a property of lower semicontinuity. The aforementioned bijection is proved to be independent of the t-norm and it provides a representation theorem for a large class of fuzzy pseudometric spaces. Further, the relations between the uniformities and topologies both generated by the fuzzy pseudometric and by the corresponding family of ordinary pseudometrics are also investigated.