CHARACTERIZATIONS OF (L;M)-FUZZY TOPOLOGY DEGREES

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.