Connectedness, Products of Finite Topologies

Question

I will show that for all $ i\in I$ (finite), $X_{i}$ is connected implies that $\prod_{i\in I} X_{i}$ is connected.

I have already shown that $X_{1}$ and $X_{2}$ are connected.

Now I am supposed to use the mathematical induction, but I am not sure how to do.

How can we show the following:

Suppose that $\prod_{i=1}^{n} X_{i}$ is connected. Is $\prod_{i=1}^{n+1}X_{i}$ also connected?

Answer 1

If you can show that the product of two connected spaces is connected, then the result follows. This is because, in the inductive step, $\prod_{i=1}^{n+1}X_{i}$ is homeomorphic to $\left(\prod_{i=1}^{n}X_{i}\right)\times X_{n+1}$.