Resumen de Fuzzy metric spaces and applications to perceptual colour-differences
Fuzzy mathematics has constituted a wide field of research, since L. A. Zadeh introduced in 1965 the concept of fuzzy set. In particular, the problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several authors. In 1994, George and Veeramani introduced and studied a notion of fuzzy metric space that constituted a modification of the one given by Kramosil and Michalek. Several authors have contributed to the study of this kind of fuzzy metrics, from the mathematical point of view and for their applications. In this thesis we have contributed to develop the study of these fuzzy metrics, from the mathematical point of view, and we approached the problem of measuring perceptual colour-difference between samples of colour using one of these fuzzy metrics. The contributions of the study carried out in this thesis is summarized as follows: \begin{enumerate} \item[(i)] We have made a detailed study of the fuzzy metric space $(X,M,\cdot)$ where $M$ is given on $X=[0,\infty[$ by $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ and others related to it. As a consequence we have introduced five questions in fuzzy metrics related to continuity, extension, contractivity and completion. \item[(ii)] We have answered an open question constructing a fuzzy metric space $(X,M,\ast)$ in which the assignment $f(t)=\lim_n M(a_n,b_n,t)$, where $\{a_n\}$ and $\{b_n\}$ are $M$-Cauchy sequences in $X$, is not a continuous function on $t$. The response to this question has allowed us to characterize the class of completable strong fuzzy metric spaces. \item[(iii)] We have introduced and studied a stronger concept than convergence of sequences in fuzzy metric spaces, which we call $s$-convergence. In our study, we have gotten a characterization of those spaces in which every convergent sequence is $s$-convergent and we have given a classification of fuzzy metrics attending to the behaviour of the fuzzy metric with respect to the different types of convergence. \item[(iv)] We have studied, in the context of fuzzy metric spaces, when certain families of open balls centered at a point are local bases for this point. \item[(v)] We have answered two open questions related to standard convergence, a stronger concept than convergence of sequences in fuzzy metric spaces, introduced in a natural way attending to the concept of standard Cauchy sequence (introduced in \cite{adomain}). These responses have led us to establish conditions under which Cauchyness and convergence should be considered \textit{compatible}. \item[(vi)] As a practical application, we have shown that a certain fuzzy metric is useful for measuring perceptual colour-differences between colour samples. \end{enumerate}
