Let $H$ denote the Hawaiian earring:
We defined a space $X$ to be semilocally simply connected if every point in $X$ has a nbhd. $U$ for which the homomorphism from the fundamental group of $U$ to the fundamental group of $X$, induced by the inclusion map, is trivial.
I'm looking for intuition on why $H$ is not semilocally simply connected.
Consider any neighborhood of the point where the circles accumulate. At least one of the circles is completely contained in that neighborhood. A loop going around that circle is a non-trivial representative of the fundamental group of $U$ and it gets mapped by the inclusion to a non-trivial loop in $X$.
To show that these loops are non-trivial consider its image by the map $X\to S^1$ that collapses all but the circle in question to a single point. This image gives us the loop that goes around $S^1$ once, which is non-trivial in $\pi(S^1)$.
