# On $\mathcal{B}$-Open Sets

## TL;DR

This paper introduces and explores the properties of $\\mathcal{B}$-open sets, a new generalization in topology, along with related concepts like $\\mathcal{B}$-density and $\\mathcal{B}$-continuity, and defines a bi-operator topological space.

## Contribution

It defines $\\mathcal{B}$-open sets as a new generalization and develops associated topological concepts and a bi-operator space framework.

## Key findings

- $\\mathcal{B}$-open sets generalize existing open set concepts.
- Introduces $\\mathcal{B}$-dense, $\\mathcal{B}$-Frechet, and contra-$\\\mathcal{B}$-closed graph.
- Defines a bi-operator topological space with two operators $T_1$ and $T_2$.

## Abstract

The aim of this paper is to define and study $\mathcal{B}$-open sets and related properties. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is in turn a generalization of a pre-open set and a semi-open set. Using $\mathcal{B}$-open sets, we introduce a number of concepts such as $\mathcal{B}$-dense, $\mathcal{B}$-Frechet, contra-$\mathcal{B}$-closed graph and contra-$\mathcal{B}$-continuity. Also, we define a bi-operator topological space $(X, \tau, T_1, T_2)$ which involves two operators $T_1$ and $T_2$, which are used to define $\mathcal{B}$-open sets.