The shrinking wedge of circles $X$, is the union of the circles $C_n$, $n=1,2,3,\dots$, where $C_n$ is the circle of radius $1/n$ and center $(1/n,0)$. I want to show that the first cohomology $H^1(X;\Bbb Z)$ is uncountable.
It is well-known that there is a surjective homomorphism $\pi_1(X)\to \Bbb Z^\Bbb N$, where $\Bbb Z^\Bbb N$ is the direct product of infinitely many copies of $\Bbb Z$, which is an uncountable group. Since $H_1(X)$ is the abelianization of $\pi_1(X)$, we also have a surjection $H_1(X)\to \Bbb Z^\Bbb N$, so $H_1(X)$ is also uncountable. Can we conclude from here that $H^1(X;\Bbb Z)$ is also uncountable? I tried to use the universal coefficient theorem, but it doesn't work so well, I think.