I have a question arising from chapter 3, page 41, in Switzer. He says "Note that every homotopy equivalence (in $\mathscr{T}$ [this is the category of topological spaces]) is a weak homotopy equivalence." This statement suggests that there are topological spaces which are weakly homotopy equivalent but not homotopy equivalent. Can anyone give an example of two such spaces?
Also, you could infer from his statement that in some other category there might be homotopy equivalences which are not weak homotopy equivalences. Is this true? That is, are there categories such that H.E. $\not\Rightarrow$ W.H.E.?
Best Answer
First note that by Whitehead's Theorem, if you know that $X$ and $Y$ are CW complexes, then "weakly homotopy equivalent" and "homotopy equivalent" coincide.
On the other hand, for general topological spaces, there are counterexamples where the two notions don't coincide.
The Long Line is not contractible, and yet all its homotopy groups vanish. So, the inclusion of a point into the long line induces a weak homotopy equivalence between them (trivially), yet they are not homotopy equivalent.