Notes:Homotopy terminology/Terminology

Terminology

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

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