Homotopic, Same homotopy type, homotopy equivalent (Are they synonyms?)
Mia Lopez 
Just to check my understanding: Are the terms
Homotopic
Same homotopy type
Homotopy equivalent
exactly the same thing?
From the texts I read, "homotopic" is often used to describe maps, but I have seen it being used to describe two spaces too, e.g. "$X$ is homotopic to $Y$".
Thanks.
$\endgroup$ 21 Answer
$\begingroup$I guess you could say that homotopic is a slightly less specific notion. We say that two maps $f,g:X\to Y$ are homotopic if there exists a so-called homotopy $F:X\times I\to Y$ so that $F(x,0)=f(x)$ and $F(x,1)=g(x)$. We usually denote this $f\simeq g$.
We can extend this notion to topological spaces $X,Y$ by saying that $X\simeq Y$ ($X$ is homotopy equivalent to $Y$) if there exist maps $f:X\to Y$ and $g:Y\to X$ so that $f\circ g\simeq Id_Y$ and $g\circ f\simeq Id_X$. One might also say that $X$ and $Y$ are homotopic (as spaces). This is not usually a source of confusion. Finally, homotopy type usually describes the same thing. That is, spaces with the same homotopy type are homotopy equivalent. For instance, $\mathbf{D}^n$ (the unit disk in $\mathbf{R}^n$) has the homotopy type of a point, i.e. $\mathbf{D}^n\simeq *$.
Note: Maps are assumed to be continuous maps of topological spaces.
$\endgroup$
