# Fundamental group of wedge sum of two coni over Hawaiian earrings

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?