Fuzzy topology with $ \mathcal{I} $-convergence

Abstract

In this paper, a new approach to specifying the closure of subsets in $ J^{E} $ is proposed, based on the concept of $ \mathcal{I} $-convergence of sequences. The notion of fuzzy $ \mathcal{I} $-closure is defined using this convergence, and from it, the fuzzy $ \mathcal{I} $-topology is introduced. Moreover, the concepts of continuity and $ \mathcal{I} $-continuity in fuzzy topological spaces are examined. A comparison between the proposed $ \mathcal{I} $-topology and the classical fuzzy topology is presented, highlighting certain fundamental properties and structural differences. This work contributes to the generalization of fuzzy topological structures by extending the role of sequence convergence through the use of $ \mathcal{I} $-convergence. Furthermore, a practical example illustrating temperature control in an industrial furnace is provided through the application of $ \mathcal{I} $-convergence of fuzzy sequences.