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What is an example of a non-contractible space $X$ with $\pi_n(X) = 0$ for all $n\geq 0$ (note in particular $X$ is path connected)?

Motivation: Whitehead's theorem implies that no such CW complex $X$ exists. I'd like to know a counterexample to the "general Whitehead theorem".

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    $\begingroup$ The closed topologists's sine curve, also known as the Warsaw circle $\endgroup$
    – Balarka Sen
    Feb 16, 2018 at 11:11
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    $\begingroup$ @BalarkaSen ah, of course. I was hoping for something more exotic. I guess the next question is to ask for locally path connected. $\endgroup$
    – Tim kinsella
    Feb 16, 2018 at 11:26
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    $\begingroup$ @Timkinsella: You should then modify your question so it asks what you actually want to ask. (An even more challenging question would be a space which is locally contractible, weakly contractible but not contractible.) $\endgroup$
    – Moishe Kohan
    Feb 17, 2018 at 15:54

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A nice example is the (open) long line. Every compact subset of it is contained in a bounded interval which is homeomorphic to $[0,1]$, and thus the homotopy groups are trivial (and the space is additionally locally contractible, even locally Euclidean!). However, it is not contractible, essentially because it's "too long" to contract the whole thing with a single interval.

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  • $\begingroup$ that is the level of exoticism I was looking for. And I think it answers the question @MoisheCohen poses in his comment. $\endgroup$
    – Tim kinsella
    Feb 18, 2018 at 2:05
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    $\begingroup$ @Timkinsella: Except I had in mind metrizable spaces... $\endgroup$
    – Moishe Kohan
    Feb 18, 2018 at 6:19
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You can find many examples among finite topological spaces. For information on these, you can browse Jonathan Barmak's LNM book in the subject.

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    $\begingroup$ I would add for those unfamiliar with finite topological spaces that every finite space is locally contractible. $\endgroup$
    – Eric Wofsey
    Feb 18, 2018 at 2:23
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Quasi-circle, which is defined in exercise 7 of page 79 in Hatcher’s book Algebraic topology

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