Notes:Homotopy terminology/Terminology

Terminology

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces. Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. Let [ilmath]C^0(X,Y)[/ilmath] denote the set of all continuous maps of the form [ilmath](:X\rightarrow Y)[/ilmath]. Let [ilmath]I:=[0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\}\subset\mathbb{R}[/ilmath]

1. Homotopy - any continuous map of the form [ilmath]H:X\times I\rightarrow Y[/ilmath] such that: [ilmath]\forall a\in A\forall s,t\in I[H(a,t)=H(a,s)][/ilmath].
   • The elements of the family [ilmath]\{h_t\}_{t\in I}[/ilmath] (where [ilmath]h_t:X\rightarrow Y[/ilmath] by [ilmath]h_t:x\mapsto H(x,t)[/ilmath]) are called the "stages" of the homotopy, [ilmath]h_0[/ilmath] is the initial stage, [ilmath]h_1[/ilmath] is the final stage
2. Homotopic - a relation on maps [ilmath]f,g\in C^0(X,Y)[/ilmath]. We write [ilmath]f\simeq g\ (\text{rel }A)[/ilmath] if there exists a homotopy ([ilmath]\text{rel }A[/ilmath])whose initial stage is [ilmath]f[/ilmath] and whose final stage is [ilmath]g[/ilmath]