Doctrine:Homotopy terminology

Terminology

Before we can define terms, here are the definitions we work with:

• Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be continuous spaces
• Let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$
• Let $C^0(X,Y)$ denote the set of continuous maps between $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$
• Let $f,g,h\in C^0(X,Y)$ be continuous maps of the form $f,g,h:X\rightarrow Y$

Terms

1. Homotopy $\mathbf{(\text{rel }A)}$ - 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 homotopy is fixed on $A$.
     • Note: if $A=$$\emptyset$ then this represents no constraint, it is vacuously true
2. Stages of a homotopy