{
  "id": "2308.06523",
  "version": "v1",
  "published": "2023-08-12T10:39:09.000Z",
  "updated": "2023-08-12T10:39:09.000Z",
  "title": "Homotopic subsets of continuous functions and their applications",
  "authors": [
    "Ali Taherifar"
  ],
  "categories": [
    "math.GN",
    "math.AT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-12T10:39:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous functions",
      "applications",
      "objects represent subsets",
      "define homotopic groups",
      "composition law governs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}