HomotopyPage

Definition

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces and let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$. A homotopy, relative to $A$ is, in its purest form, is any continuous map:

• $F:X\times I\rightarrow Y$ (where $I:=[0,1]\subset\mathbb{R}$ - the unit interval) such that $\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]$

Stages of the homotopy

Let $t\in I$ be given, and $H:X\times I\rightarrow Y$ be a homotopy as defined above. $h_t:X\rightarrow Y$ denotes a stage of the homotopy and is defined as follows:

• $h_t:x\mapsto H(x,t)$

The family of stages, $\{h_t:X\rightarrow Y\}_{t\in I}$, are collectively called the stages of the homotopy and

1. $h_0:X\rightarrow Y$ defined by $h_0:x\mapsto H(x,0)$ is the initial stage of the homotopy.
2. $h_1:X\rightarrow Y$ defined by $h_1:x\mapsto H(x,1)$ is the final stage of the homotopy.

Note that the stages of a homotopy are continuous

Purpose

The point of a homotopy is to be a relation (in fact an equivalence relation) of (continuous) maps.

Homotopic maps

Homotopic maps/Definition