On sparse set topology using ideals in the space of reals

TL;DR

This paper introduces the concept of $\\mathcal{I}$-sparse sets in real spaces, defines a new topology based on these sets, and compares its properties to existing topologies, revealing it to be finer than the $\mathcal{I}$-density topology.

Contribution

It proposes a novel topology called $\mathcal{I}$-sparse set topology and analyzes its properties, expanding the understanding of set topologies in real spaces.

Findings

The $\mathcal{I}$-sparse set topology is finer than the $\mathcal{I}$-density topology.

Properties of $\mathcal{I}$-sparse sets are characterized and studied.

The topology exhibits distinct features compared to existing topologies in the space of reals.

Abstract

In this paper we have introduced the notion of $\mathcal{I}$-sparse set in the space of reals and explored some properties of the family of $\mathcal{I}$-sparse sets. Thereafter we have induced a topology namely $\mathcal{I}$-sparse set topology in the space of reals and it has been observed that this topology is finer than $\mathcal{I}-$density topology introduced by Banerjee and Debnath in \cite{banerjee 4}. We further studied some salient properties of this topology.