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On sparse set topology using ideals in the space of reals

Indrajit Debnath, Amar Kumar Banerjee

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.

On sparse set topology using ideals in the space of reals

Abstract

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

Paper Structure

This paper contains 4 sections, 24 theorems, 28 equations.

Table of Contents

  1. Introduction and preliminaries
  2. $\mathcal{I}$-density
  3. $\mathcal{I}$-Sparse set
  4. $\mathcal{I}$-Sparse set topology

Key Result

Theorem 1.1

Oxtoby For any set $H \in \mathcal{L}$,

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.5
  • Proposition 1.6
  • Example 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • Definition 2.1
  • ...and 43 more