A classic example in algebraic topology is The Shrinking Wedge of Circles $X$, which is the union of circles $C_n$ of radius $\frac{1}{n}$ and center $(\frac{1}{n},0 )$ for $n=1,2,3,…$.
In Hatcher’s, it says that the product of surjections $\rho_n:\pi_1 (X) \to \pi _1(C_n)$ gives a surjective homomorphism $\rho:\pi_1(X) \to \prod_\infty Z$. And since $\prod_\infty Z$ is uncountable, $\pi_1(X)$ is uncountable.
I can’t understand why $\prod_\infty Z$ is uncountable? Why is the direct product of countable groups uncountable?