If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X = $ (the Hawaiian Earring) which collapses the segment {0} x I to a point, show that $\pi_1(C)$ is uncountable by constructing a homomorphism from $\pi_1(C)$ onto $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$.
From an example in the book, I know there is a surjection from the Hawaiian Earring, AKA $\sum X$ in this case, to $\prod_\infty \mathbb{Z}$. I also know that the fundamental group of SX is countable from an earlier part of the problem. It seems as if any single loop in $\pi_1(C)$ might be retracted through the collapsed point of the mapping cone $(0, X)$ into a constant path, but it's extremely difficult for me to visualize it. It certainly seems like the only way for a surjection like the one requested to exist is if something like that works for every finite collection of loops, but somehow doesn't work for any infinite collection.
Any help at all understanding the nature of the proposed surjection would be appreciated.
Hint: let $U=C(SX)$, the cone over $S(X)$, and let $V=S X\cup_f \Sigma X\times [0,1]$ where $f\colon S X\to \Sigma X\times [0,1]$ is given by $f(x)=(q(x),1)$ where $q\colon SX\to\Sigma X$ is the canonical quotient map.
If we glue $U$ to $V$ in the obvious way along their embedded copies of $SX$ then we get a space homeomorphic to the mapping cone $C$. Apply the Van Kampen Theorem to these two pieces of the mapping cone, and recall that a cone over any space is simply connected.