^{1}School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, P.R. China

^{2}School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, P.R. China and Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, P.R. China

Abstract

In this paper, characterizations of the degree to which a mapping $\mathcal{T} : L^{X}\longrightarrow M$ is an $(L, M)$-fuzzy topology are studied in detail. What is more, the degree to which an $L$-subset is an $L$-open set with respect to $\mathcal{T}$ is introduced. Based on that, the degrees to which a mapping $f: X\longrightarrow Y$ is continuous, open, closed or a quotient mapping with respect to $\mathcal{T}_{X}$ and $\mathcal{T}_{Y}$ are defined, and their characterizations are given, respectively. Besides, the relationships among the continuity degrees, the openness degrees, the closedness degrees and the quotient degrees of mappings are discussed.