The main purpose of this article is to introduce and study some new spaces of I-convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operator i.e 3SI (μ,ν)(T ) and 3SI0(μ,ν)(T ) and examine some fundamental properties, fuzzy topology and verify inclusion relations lying under these spaces.

A class with ambiguous status of its elements by a membership function that assigns to each element a grade in the close interval [ 0 , 1 ] ; Lofti Zadeh introduced this idea into theory as fuzzy sets in the year of 1965. A study of fuzzy anti-normed linear spaces by Kočinac on some topological properties motivated us to work on fuzzy anti-normed triple sequence spaces with respect to ideal by using compact linear operator. Further, we prove some theorems, particularly on convergence and completeness.

In the final years of the 20th century, the notion of generalized topological spaces was introduced, marking a significant shift in the field of topology. This paper focuses on a subset of on a non-empty set that is closed under arbitrary unions, defining a generalized topology and subsequently a generalized topological space (GTS) denoted by . Within this framework, we explore the concept of Noetherian generalized topological spaces and delve into the properties of closed subsets within the Noetherian GTS. The investigation reveals that subspaces of a Noetherian GTS , with the induced topology, inherit the Noetherian property and exhibit finitely many non-empty irreducible components. Furthermore, the study extends to the analysis of hereditary properties, regular , , irreducible closed subsets, and the product properties of closed subsets under continuous functions. We also establish the closure property of finite unions in Noetherian GTS and clarify the homeomorphic nature of Noetherian GTS to itself.

The purpose of the present article is to introduce the idea of rough ideal statistical convergence of order $\alpha$ of sequences in intuitionistic fuzzy normed spaces. In these spaces, we have defined the concepts of rough ideal statistical limit points, rough ideal cluster points, and rough ideal boundedness. The theory of convexity and closeness was next examined for the collection of approximative statistical limit points.

The concept of fuzzy sets was introduced by Zadeh as a means of representing data that was not precise but rather fuzzy. Recently, Kocinac [19] studied some topological properties of fuzzy antinormed linear spaces. This has motivated us to introduce and study the fuzzy antinormed double sequence spaces with respect to ideal by using a bounded linear operator and prove some theorems, in particular convergence and completeness theorems on these new double sequence spaces.

In the present work, we introduce paranormed zweier ideal convergent triple sequence spaces defined by a compact operator 3ZI(?), 3ZI0 (?) and 3ZI? (?) where q = (qijk) is a triple sequence of positive numbers and we study some algebraic and topological properties of these spaces.