Homotopic maps
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]f,g:X\rightarrow Y[/ilmath] be continuous maps and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath].
 We say "[ilmath]f[/ilmath] is homotopic to [ilmath]g[/ilmath] (relative to [ilmath]A[/ilmath])" if there exists a homotopy [ilmath](\text{rel }A)[/ilmath]^{[Note 1]} whose initial stage is [ilmath]f[/ilmath] and whose final stage is [ilmath]g[/ilmath].
 This is written: [ilmath]f\simeq g\ (\text{rel}\ A)[/ilmath]
 or simply [ilmath]f\simeq g[/ilmath] if [ilmath]A=\emptyset[/ilmath]
 If [ilmath]A=\emptyset[/ilmath] (and we write [ilmath]f\simeq g[/ilmath]) we may say that [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are freely homotopic
 This is written: [ilmath]f\simeq g\ (\text{rel}\ A)[/ilmath]
 The homotopy [ilmath](\text{rel }A)[/ilmath] that exists if [ilmath]f\simeq g\ (\text{rel }A)[/ilmath], say [ilmath]F:X\times I\rightarrow Y[/ilmath], with [ilmath]\forall x\in X[(F(x,0)=f(x))\wedge(F(x,1)=g(x))][/ilmath] and [ilmath]\forall a\in A\forall t\in I[F(a,t)=f(a)=g(a)][/ilmath], is called a homotopy of maps
We say [ilmath]f\simeq g\ (\text{rel }A)[/ilmath] (or [ilmath]f\simeq g[/ilmath] if [ilmath]A=\emptyset[/ilmath]) if:
 There exists a continuous map, [ilmath]F:X\times I\rightarrow Y[/ilmath] (a homotopy) such that:
 [ilmath]\forall x\in X[F(x,0)=f(x)][/ilmath]  the initial stage of the homotopy is [ilmath]f[/ilmath]
 [ilmath]\forall x\in X[F(x,1)=g(x)][/ilmath]  the final stage of the homotopy is [ilmath]g[/ilmath]
 [ilmath]\forall a\in A\forall s,t\in I[F(a,s)=F(a,t)][/ilmath]^{[Note 2]}  or equivalently  [ilmath]\forall a\in A\forall t\in I[F(a,t)=f(x)=g(x)][/ilmath]  the homotopy is fixed on [ilmath]A[/ilmath]
We can use this to define a relation on continuous maps:
 If [ilmath]f\simeq g\ (\text{rel }A)[/ilmath] then we consider [ilmath]f[/ilmath] and [ilmath]g[/ilmath] related and say "[ilmath]f[/ilmath] is homotopic to [ilmath]g[/ilmath] ([ilmath]\text{rel }A[/ilmath])"
Claim: this is an equivalence relation (see: the relation of maps being homotopic is an equivalence relation)
Notes
 ↑ Recall a homotopy (relative to [ilmath]A[/ilmath]) is a continuous map, [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath]  the unit interval) such that:
 [ilmath]\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)][/ilmath]
 ↑ Note that if [ilmath]A=\emptyset[/ilmath] then this represents no condition/constraint on [ilmath]F[/ilmath], as are not any [ilmath]a\in A[/ilmath] for this to be true on!
References
OLD PAGE
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces. Let [ilmath]f,g:X\rightarrow Y[/ilmath] be continuous maps. The maps [ilmath]f[/ilmath] and [ilmath]h[/ilmath] are said to be homotopic^{[1]} if:
 there exists a homotopy, [ilmath]H:X\times I\rightarrow Y[/ilmath], such that [ilmath]H_0=f[/ilmath] and [ilmath]H_1=g[/ilmath]  here [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] denotes the unit interval.
 (Recall for [ilmath]t\in I[/ilmath] that [ilmath]H_t:X\rightarrow Y[/ilmath] (which denotes a stage of the homotopy) is given by [ilmath]H_t:x\mapsto H(x,t)[/ilmath])
TODO: Mention freehomotopy, warn against using null (as that term is used for loops, mention relative homotopy
See also
 Homotopy  any continuous map of the form [ilmath]H:X\times I\rightarrow Y[/ilmath]
 Homotopy is an equivalence relation
 Pathhomotopy
 Fundamental group
References
Template:Homotopy theory navbox
