# Are there any spaces with vanishing homotopy groups but not contractible? [closed]

I'm studying basic Algebraic Topology. Here is my question:

Let $X$ be a space with $\pi_i(X)=0$ for all $i$.

Then, $X$ should be contractible?

(If $X$ is CW-complex, then cell-by cell argument works and this is true.)

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## closed as too localized by Mark Grant, Gjergji Zaimi, Ryan Budney, Loop Space, Andreas ThomDec 7 '11 at 12:44

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See the "quasi-circle" example in Hatcher, exercise 7 section 1.3. It has trivial homotopy groups but it's not contractible. – Ryan Budney Dec 7 '11 at 8:24
This type of question is probably best put on math.stackexchange.com, as its a common example in introductory algebraic topology courses. – Ryan Budney Dec 7 '11 at 9:07