Is $\pi_n(\Pi_\alpha X_\alpha) \approx \Pi_\alpha \pi_n(X_\alpha)$ if the $X_\alpha$ aren't path connected?

Question

Consider the following proposition from Hatcher's Algebraic Topology:

Is it true that $\pi_n(\Pi_\alpha X_\alpha) \approx \Pi_\alpha \pi_n(X_\alpha)$ if the $X_\alpha$ aren't path connected? Is there a counterexample? Or is the proof very different?

Answer 1

It's more that $\pi_n(X_\alpha)$ is not defined without choosing a basepoint. If $x_\alpha \in X_\alpha$, and $x \in \Pi_\alpha X_\alpha$ is the point whose $\alpha$ coordinate is $x_\alpha$, then $$\pi_n(\Pi_\alpha X_\alpha, x) = \Pi_\alpha \pi_n(X_\alpha, x_\alpha) .$$ This follows from the quoted proposition by restricting to the path component of $X_\alpha$ that contains $x_\alpha$.