Homotopy

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Definition

A homotopy between two topological spaces, $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$, is a continuous function:

• $F:X\times I\rightarrow Y$ (where $I$ denotes the unit interval, $[0,1]\subset\mathbb{R}$)

A homotopy is relative to $A\in\mathcal{P}(X)$ if $F(a,t)$ is independent of $t$ for all $a\in A$

Terminology

The family of functions $\{f_t:X\rightarrow Y\ \vert\ \forall t\in[0,1],\ f_t:x\mapsto F(x,t)\}$ are called the stages of the homotopy. So we might say:

• Let $f_t$ be a stage of the homotopy $F$ or something similar

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Definition

A homotopy from the topological spaces $(X,\mathcal{ J })$ to $(Y,\mathcal{ K })$ is a continuous function:

• $F:X\times I\rightarrow Y$ (where $I$ denotes the unit interval, $[0,1]\subseteq\mathbb{R}$)

For each $t\in I$ we have a function:

• $F_t:X\rightarrow Y$ defined by $F_t:x\mapsto F(x,t)$ - these functions, the $F_t$ are called the stages of the homotopy.

Applications

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Homotopic maps

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Conditions

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Relative homotopic maps

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Conditions

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Path homotopy

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Conditions