In the paper, we analyze spaces with fuzzy sets through the perspective of the theory of operators over these sets. We consider spaces with pretopological, topological, and partition structures, and we show that they are images of Čech and Kuratowski interior operators. In particular, the Kuratowski interior operators can be expressed using an \({\textsf{L}}\) -fuzzy relation, which is reflexive. We focus particularly on the weakest \({\textsf{L}}\) -fuzzy partitioned space. Subsequently, we prove that an \({\textsf{L}}\) -fuzzy lower \({\textsf{F}}^{\downarrow }\) -transform (resp. upper \({\textsf{F}}^{\uparrow }\) -transform) is a strong Čech–Alexandroff \({\textsf{L}}\) -fuzzy interior operator ( \({\textsf{L}}\) -FIO) (resp. \({\textsf{L}}\) -fuzzy closure operator ( \({\textsf{L}}\) -FCO)). Conversely, we discovered circumstances which ensure that every strong Alexandroff \({\textsf{L}}\) -fuzzy pretopology (resp. co-pretopology) can be generated by an \({\textsf{L}}\) -valued lower \({\textsf{F}}^{\downarrow }\) -transform (resp. upper \({\textsf{F}}^{\uparrow }\) -transform). As a new methodological tool, we use the theory of \({\textsf{L}}\) -fuzzy relation equations ( \({\textsf{L}}\) -FREs) and especially conditions for their solvability.
