Proposition 1.12 from Hatcher's Algebraic Topology book states that \begin{align*} \pi_1(X \times Y) \; \text{is isomorphic to} \; \pi_1(X) \times \pi_1(Y) \; \text{if X and Y are path connected} \; \end{align*} I tried to understand the proof but I fail to see where the condition of path connectedness is needed, aside from the fact that one had to fix a base point otherwise. So I guess my question is:
Is $\pi_1(X \times Y, (x_o,y_0))$ isomorphic to $\pi_1(X,x_0) \times \pi_1(Y,y_0)$ even for spaces that are not path connected? If not, why not and is there an easy counter example?
Thanks in advance