Connectedness, local connectedness, and components on bipolar soft generalized topological spaces

Abstract

Connectedness represents the most significant and fundamental topological property. It highlights the main characteristics of topological spaces and distinguishes one topology from another. There is a constant study of bipolar soft generalized topological spaces (\( \mathcal{BSGTS}s \)) by presenting \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-connected set and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-connected space in \(\mathcal{BSGTS}s\) as well as it is discussing some properties and results for these topics. Additionally, the notion of bipolar soft disjoint sets is put forward, \(\mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-separation set, \(\widetilde{\widetilde{\mathfrak{g}}}\)-separated \(\mathcal{BSS}s\) and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-hereditary property. Moreover, there is an extensive study of \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-locally connected space and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-component with some related properties and theorems following them, such as the concepts of \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-locally connected spaces and \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-connected are independent of each other; also determined the conditions under which the \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-connected subsets are \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-components.