A Novel Framework for Bioperations on $e^{\star}$-Open Sets: Theoretical Foundations and Share Market Applications

Bioperations on $e^\star$-Open Sets

Abstract

In this paper, we introduce the concept of $e^\star_{[\gamma,\gamma']}$-open sets in topological spaces and examine their properties in detail. Additionally, we propose a new class of functions, termed $(e^\star_{[\gamma,\gamma']}, e^\star_{[\beta,\beta']})$-continuous functions, and explore their fundamental characteristics. To demonstrate the practical relevance of these theoretical constructs, we provide an illustrative example showing how $e^\star_{[\gamma,\gamma']}$-open sets can be applied to identify stable financial patterns in the share market. This application bridges abstract topological theory with real-world market dynamics, offering a novel perspective on analyzing stock price movements and trader behavior.