Generalization of continuity

Articles in Press

Document Type : Research Paper

Authors

1 Faculty of Mathematics and Computer Science, Hakim Sabzevari University, Sabzevar, Iran

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

10.48308/cgasa.2026.243504.1604

Abstract

This paper aims to redefine the concept of ``continuity of a function at a point'' from a set-theoretic perspective, providing a sufficiently flexible definition that encompasses the various forms of continuity found in the mathematical literature. Let $\mathscr{A}$ and $\mathscr{B}$ denote families of subsets of non-empty sets $X$ and $Y$, respectively. We define an $\mathscr{A}$-$ \mathscr{B}$-continuous map and examine some algebraic properties of structures related to the set $C_{(\mathscr{A}, \mathscr{B})}(X, Y)$, which consists of all $\mathscr{A}$-$ \mathscr{B}$-continuous maps from $X$ to $Y$. Additionally, we show that $C_{_{(\mathscr{A}, \mathcal{O}Y)}}(X, Y)\cong C(X_z, Y)$, where $Y$ is an $f$-ring and $\mathscr{A}$ is closed under finite intersections, and $X_z$ is a topological space induced by $(X, \mathscr{A})$. 

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References

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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 06 June 2026

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