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Probabilistic uniformities of uniform spaces

Last modified: 01-10-2019

#### Abstract

The theory of metric spaces in the fuzzy context has shown to be an interesting area of study not only from a theoretical point of view but also for its applications. Nevertheless, it is usual to consider these spaces as classical topological or uniform spaces and there are not too many results about constructing fuzzy topological structures starting from a fuzzy metric. Maybe, H\"ohle was the first to show how to construct a probabilistic uniformity and a Lowen uniformity from a probabilistic pseudometric \cite{Hohle78,Hohle82a}. His method can be directly translated to the context of fuzzy metrics and allows to characterize the categories of probabilistic uniform spaces or Lowen uniform spaces by means of certain families of fuzzy pseudometrics \cite{RL}.

On the other hand, other different fuzzy uniformities can be constructed in a fuzzy metric space: a Hutton $[0,1]$-quasi-uniformity \cite{GGPV06}; a fuzzifiying uniformity \cite{YueShi10}, etc. The paper \cite{GGRLRo} gives a study of several methods of endowing a fuzzy pseudometric space with a probabilistic uniformity and a Hutton $[0,1]$-quasi-uniformity.

In 2010, J. Guti\'errez Garc\'{\i}a, S. Romaguera and M. Sanchis \cite{GGRoSanchis10} proved that the category of uniform spaces is isomorphic to a category formed by sets endowed with a fuzzy uniform structure, i. e. a family of fuzzy pseudometrics satisfying certain conditions. We will show here that, by means of this isomorphism, we can obtain several methods to endow a uniform space with a probabilistic uniformity. Furthermore, these constructions allow to obtain a factorization of some functors introduced in \cite{GGRoSanchis10}.

\section*{Acknowledgements} % If any

The first and third authors are supported by the grant MTM2015-64373-P (MINECO/FEDER, UE).

\begin{thebibliography}{99}

\footnotesize

\bibitem{GGPV06}

J.~Guti\'errez-Garc\'{\i}a and M.~A. {de Prada Vicente}, Hutton [0, 1]-quasi-uniformities induced by fuzzy (quasi-)metric

spaces, Fuzzy Sets Syst. 157 (2006), no.~6, 755--766.

\bibitem{GGRLRo}

J.~Guti\'errez-Garc\'{\i}a, J.~Rodr\'{\i}guez-L\'opez and S.~Romaguera,

Fuzzy uniformities of fuzzy metric spaces, Fuzzy Sets Syst., to

appear.

\bibitem{GGRoSanchis10}

J.~Guti\'errez-Garc\'{\i}a, S.~Romaguera and M.~Sanchis, Fuzzy uniform

structures and continuous t-norms, Fuzzy Sets Syst. 161 (2010),

no.~7, 1011--1021.

\bibitem{Hohle78}

U.~H\"ohle, Probabilistic uniformization of fuzzy topologies, Fuzzy Sets

Syst. 1 (1978), 311--332.

\bibitem{Hohle82a}

U.~H\"ohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets

Syst. 8 (1982), 63--69.

\bibitem{RL}

J.~Rodr\'{\i}guez-L\'opez, Fuzzy uniform structures, Filomat, to appear.

\bibitem{YueShi10}

Y.~Yue and F.-G. Shi, On fuzzy pseudo-metric spaces, Fuzzy Sets Syst. 161

(2010), 1105--1116.

\end{thebibliography}

On the other hand, other different fuzzy uniformities can be constructed in a fuzzy metric space: a Hutton $[0,1]$-quasi-uniformity \cite{GGPV06}; a fuzzifiying uniformity \cite{YueShi10}, etc. The paper \cite{GGRLRo} gives a study of several methods of endowing a fuzzy pseudometric space with a probabilistic uniformity and a Hutton $[0,1]$-quasi-uniformity.

In 2010, J. Guti\'errez Garc\'{\i}a, S. Romaguera and M. Sanchis \cite{GGRoSanchis10} proved that the category of uniform spaces is isomorphic to a category formed by sets endowed with a fuzzy uniform structure, i. e. a family of fuzzy pseudometrics satisfying certain conditions. We will show here that, by means of this isomorphism, we can obtain several methods to endow a uniform space with a probabilistic uniformity. Furthermore, these constructions allow to obtain a factorization of some functors introduced in \cite{GGRoSanchis10}.

\section*{Acknowledgements} % If any

The first and third authors are supported by the grant MTM2015-64373-P (MINECO/FEDER, UE).

\begin{thebibliography}{99}

\footnotesize

\bibitem{GGPV06}

J.~Guti\'errez-Garc\'{\i}a and M.~A. {de Prada Vicente}, Hutton [0, 1]-quasi-uniformities induced by fuzzy (quasi-)metric

spaces, Fuzzy Sets Syst. 157 (2006), no.~6, 755--766.

\bibitem{GGRLRo}

J.~Guti\'errez-Garc\'{\i}a, J.~Rodr\'{\i}guez-L\'opez and S.~Romaguera,

Fuzzy uniformities of fuzzy metric spaces, Fuzzy Sets Syst., to

appear.

\bibitem{GGRoSanchis10}

J.~Guti\'errez-Garc\'{\i}a, S.~Romaguera and M.~Sanchis, Fuzzy uniform

structures and continuous t-norms, Fuzzy Sets Syst. 161 (2010),

no.~7, 1011--1021.

\bibitem{Hohle78}

U.~H\"ohle, Probabilistic uniformization of fuzzy topologies, Fuzzy Sets

Syst. 1 (1978), 311--332.

\bibitem{Hohle82a}

U.~H\"ohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets

Syst. 8 (1982), 63--69.

\bibitem{RL}

J.~Rodr\'{\i}guez-L\'opez, Fuzzy uniform structures, Filomat, to appear.

\bibitem{YueShi10}

Y.~Yue and F.-G. Shi, On fuzzy pseudo-metric spaces, Fuzzy Sets Syst. 161

(2010), 1105--1116.

\end{thebibliography}

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