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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.?

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  • $\begingroup$ In the third sentence of the question, "This statement implies" should probably be "This statement suggests". A statement of the form "every X is Y" doesn't in general imply that the converse fails. But if someone asserts that every X is Y and doesn't assert the converse, one might reasonably suspect that the converse fails. $\endgroup$ – Andreas Blass May 24 '13 at 18:27
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    $\begingroup$ The simplest example I know of is $X = \{0\}\cup \{1/n \vert n\ge 1\} \subset \mathbb R$ as a sub space, and $Y$ a countable discrete space, then the obvious map $Y \to X$ is a weak equivalence, but you can prove that there can be no homotopy equivalence essentially because $1/n \to 0$. $\endgroup$ – Justin Young May 24 '13 at 19:43
  • $\begingroup$ No one seemed to answer your second question, but as stated the answer is 'no'. A homotopy equivalence is always a weak equivalence... I'm not sure what other categories you had in mind, but any one where you can define homotopy equivalence would be a subcategory of spaces. Unless you're asking a question about model categories in general... $\endgroup$ – Dylan Wilson May 30 '13 at 9:57
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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.

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In general, it is true that there are spaces which are weakly homotopy equivalent but not homotopy equivalent. One example would be the point and the Warsaw Circle where the unique map from the Warsaw circle to the point induces an isomorphism on all homotopy groups, and so they are weakly equivalent, but the Warsaw circle is not contractible and so not homotopy equivalent to a point.

However, in the category of CW-complexes, if $X$ and $Y$ are weakly homotopy equivalent, then they are homotopicy equivalent. This is a theorem known as Whitehead's Theorem.

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