A new perspective on fuzzy mapping theory with invertedly open and closed mappings

In fuzzy mapping theories, we examine fuzzy closedness and fuzzy continuity of a mapping $\varphi$, characterized respectively by $\overline{\varphi (M)}\subseteq \varphi (\overline{M})$ and $\varphi (\overline{M})\subseteq \overline{\varphi (M)}$, for every fuzzy set $M$ in $V$. Here, $(V,\tau )$ represents a fuzzy topological space (FTs), where $V=\{v\}$ denotes a set of points. This reveals a fundamental symmetry between the two mappings in connection with the closure operator. On the other hand, the fuzzy openness of a mapping $\varphi$ is characterized by $\varphi (M^{\circ })<(\varphi (M){)}^{\circ }$ for every fuzzy set $M$ in $V$. Considering the above statements, it is logical to explore how fuzzy continuity relates to the interior operator. Building on this, we introduce the notion of the invertedly fuzzy open mapping, defined as $(\varphi (M){)}^{\circ }<\varphi (M^{\circ })$ for any fuzzy set $M$ in $V$, and discuss its relationship with fuzzy continuity. In our study, we define and analyze invertedly fuzzy open and invertedly fuzzy closed mappings, along with their respective properties. We also delve into how these mappings connect with fuzzy continuous mappings. Furthermore, we examine a characterization of fuzzy homeomorphism for bijective mappings concerning the interior operator.