Temporal topological operators with picture fuzzy multifunctions

This paper introduces the notion of a temporal picture fuzzy modal topological structure (TPFMTS). These structures are based on novel temporal picture fuzzy topological operators as common modal topological operators standing for the closure and interior operators. The paper discusses several fundamental properties of these new TPFMTSs. The constructed temporal picture fuzzy modal topological operators establish that some properties that were considered satisfactory in the temporal intuitionistic fuzzy modal topological structures, as defined by Atanassov in 2023, are not fulfilled. Also, we introduce the concepts of $TP{F}^u$ and $TP{F}^l$-continuous and almost continuous multifunctions. Several properties and characterizations of the presented continuous multifunctions and their types of continuity are established. Some examples are given to explain the correct implications between these notions.