On certain generalized notions using $\mathcal{I}$-convergence in topological spaces

TL;DR

This paper explores generalized topological properties and mappings based on ideal convergence using a functional approach, extending classical results and introducing new concepts like -functional spaces and mappings.

Contribution

It introduces and studies -functional spaces and mappings, unifying and extending existing results through a general functional framework based on ideals.

Findings

Defined -functional spaces and mappings.

Extended classical topological results using ideal convergence.

Unified proofs for properties of various topological spaces.

Abstract

In this paper, we consider certain topological properties along with certain types of mappings on these spaces defined by the notion of ideal convergence. In order to do that, we primarily follow in the footsteps of the earlier studies of ideal convergence done by using functions (from an infinite set $S$ to $X$) in \cite{CS, das4, das5}, as that is the most general perspective and use functions instead of sequences/nets/double sequences etc. This functional approach automatically provides the most general settings for such studies and consequently extends and unifies the proofs of several old and recent results in the literature about spaces like sequential, Fr\'{e}chet-Uryshon spaces and sequential, quotient and covering maps. In particular, we introduce and investigate the notions of $\ic$-functional spaces, $\ic$-functional continuous, quotient and covering mappings and finally $\ic$-functional Fr\'{e}chet-Uryshon spaces. In doing so, we take help of certain set theoretic and other properties of ideals.