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Let $X_1$ and $X_2$ be two cones over the hawaiian earring and let $X$ be the wedge sum of $X_1$ and $X_2$ (of course you join them in the special point of the hawaiian earring). How do you prove that $X$ is not contractible? Thanks

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    $\begingroup$ @Pedro: If this is a homework, then MO is not an appropriate place for the question, try math.stackexchange instead. If this is not a homework, you should give a motivation for the question, explain why do you think the space is not contractible, what did you try to solve this problem, etc. $\endgroup$
    – Misha
    Commented Dec 21, 2012 at 18:06
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    $\begingroup$ Also: Try to search "wedge sum" on MO; the first hit you see will get you started on the problem regardless of its origin. $\endgroup$
    – Misha
    Commented Dec 21, 2012 at 18:18

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See the proof of Theorem 2.6 of

J.W. Cannon, G.R. Conner, The combinatorial structure of the Hawaiian Earring group, Topology Appl. 106 (3) (2000) 225–271.

A copy of this paper is available on the second author's webpage here.

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The space you are talking about is sometimes called the Griffiths space. As Henry Horton suggests, to prove it is not contractible you can show the fundamental group is non-trivial (even though its Cech homotopy groups are trivial). The reference already given is a good one though I'll add that Griffith was the first to show this in

H. B. Griffiths, The fundamental group of two spaces with a common point, Quart. J. Math. 5 (1954), 175-190.

Another discussion on non-contractible one-point unions of simply connected spaces is in

K. Eda, A locally simply connected space and fundamental groups of one point unions of cones, Proc. Amer. Math. Soc. 116 No. 1 (1992) 239-249.

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