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Consider a conus $C\mathbb H$ over the Hawaiian earring $\mathbb H$ with the bad point $p\in\mathbb H$. Its fundamental group is trivial.

However, I think the fundamental group of the wedge sum $C\mathbb H\vee_p C\mathbb H$ is uncountable (probably isomorphic to that of $\mathbb H\vee_p \mathbb H$; in particular, I think $C\mathbb H\vee_p C\mathbb H$ is not semilocally simply connected).

How can I prove this?

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This is called The Griffiths Twin Cone, and its fundamental group is indeed uncountable.

This space was studied in:

H.B. Griffiths, The fundamental group of two spaces with a common point, Quart. J. Math. Oxford (2) 5 (1954) 175-190.

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

A simple description can be found here.

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