Homotopic subsets of continuous functions and their applications

In this paper, we introduce the notion of bi-homotopy between subsets of continuous functions. A map $\phi$ from $A$ to $B$ is called an $h$-map if, for each two homotopic maps $f, g\in A$, their image (i.e., $\phi(f), \phi(g)$) are homotopic in $B$. We call an $h$-map $\phi$ from $A$ to $B$ a bi-homotopy if it satisfies two conditions. First, for any $f, g \in A$, $\phi(f)$ is homotopic to $\phi(g)$ in $B$ implies $f$ is homotopic to $g$ in $A$. Next, for each $g \in B$, there exists an $f \in A$ such that $\phi(f)$ is homotopic to $g$ in $B$. We establish the concept of homotopy equivalence between subsets $A$ and $B$ (denoted as $A \simeq B$) as the existence of two bi-homotopies $\phi$ from $A$ to $B$ and $\psi$ from $B$ to $A$, satisfying $\phi\psi(h)$ is homotopic to $h$ for every $h \in B$, and $\psi\phi(h)$ is homotopic to $h$ for every $h \in A$. We then apply this definition to characterize homotopic subsets of continuous functions and introduce novel categories of subsets of $C(X, Y)$, notably the category $\mathcal{P}(C(X, Y))$, where $X, Y$ are two topological spaces. In this category, objects represent subsets of $C(X, Y)$, morphisms denote bi-homotopies between any two objects, and a composition law governs the combination of morphisms. Furthermore, we extend this framework to define homotopic groups (resp., rings) of continuous functions when $Y$ is a topological group (resp., topological ring). Leveraging topological properties of $X$ and $Y$, we investigate the group (resp., ring) properties of $C(X, Y)$. We discuss potential applications and implications of the introduced bi-homotopy concept in the study of continuous functions and their subsets.