Finite Product Space is Connected iff Factors are Connected

Theorem
Let $T_1 = \left({S_1, \tau_1}\right), T_2 = \left({S_2, \tau_2}\right), \ldots, T_n = \left({S_n, \tau_n}\right)$ be metric spaces.

Let $\mathcal T = \displaystyle \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$.

Then $T$ is connected each of $T_1, T_2, \ldots, T_n$ are connected.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 2}$, let $P \left({n}\right)$ be the proposition:
 * $T$ is connected each of $T_1, T_2, \ldots, T_n$ are connected.

Basis for the Induction
$P \left({2}\right)$ is the case:
 * The product space $T_1 \times T_2$ is connected $T_1$ and $T_2$ are connected.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is the induction hypothesis:
 * $\displaystyle \prod_{i \mathop = 1}^k T_i$ is connected each of $T_1, T_2, \ldots, T_k$ are connected.

from which it is to be shown that:
 * $\displaystyle \prod_{i \mathop = 1}^{k + 1} T_i$ is connected each of $T_1, T_2, \ldots, T_{k + 1}$ are connected.

Induction Step
This is the induction step:

By definition of product space:
 * $\displaystyle \prod_{i \mathop = 1}^{k + 1} T_i = \left({\prod_{i \mathop = 1}^k T_i}\right) \times T_{k + 1}$

But from the basis for the induction:
 * $\displaystyle \prod_{i \mathop = 1}^{k + 1} T_i$ is connected $\displaystyle \prod_{i \mathop = 1}^k T_i$ is connected and $T_{k + 1}$ is connected

and from the induction hypothesis:
 * $\displaystyle \prod_{i \mathop = 1}^k T_i$ is connected each of $T_1, T_2, \ldots, T_k$ are connected.

Hence:


 * $\displaystyle \prod_{i \mathop = 1}^{k + 1} T_i$ is connected each of $T_1, T_2, \ldots, T_{k + 1}$ are connected.

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction:
 * For all $n \in \Z_{\ge 2}$, $T$ is connected each of $T_1, T_2, \ldots, T_n$ are connected.

Also see

 * Product Space is Path-connected iff Factor Spaces are Path-connected