Summary: In this paper, characterizations of the degree to which a mapping \(\mathcal{T}\,:\,L^X\to M\) 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 \(f\,:\,X\to Y\) 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.