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my notes say:

What does a rel{0,1} homotopy mean? Keeping the endpoints [itex] \alpha(0), \alpha(1) \in X[/itex] of a path [itex]\alpha : I \rightarrow X[/itex] fixed during a homotopy.

we do this anyway don't we?

and then more precisely they say:

A homotopy rel {0,1} of two paths [itex] \alpha_0 , \alpha_1 : I \rightarrow X[/itex] with the same endpoints [itex]\alpha_0(0) = \alpha_1(0) , \alpha_0 (1) = \alpha_1(1) \in X[/itex]

is a collection of paths [itex]h_t : I \rightarrow X ( t \in [0,1]) [/itex] witht he same endpoints [itex]h_t(0)=\alpha_0(0)=\alpha_1(0) , h_t(1) = \alpha_0(1) = \alpha_1(1)[/itex]

such that [itex]h_0 = \alpha_0 , h_1 = \alpha_1[/itex] and such that the function

[itex] h : I \times I \rightarrow X ; (s,t) \mapsto h_t(s)[/itex] is continuous.

I can't see much difference between this and the standard homotopy definition other than the fact that such a homotopy could only be defined for maps whose endpoints are the same to begin with.

also how does [itex]h_0 = \alpha_0 , h_1 = \alpha_1[/itex]???

thanks.