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 Cartesian product 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 Cartesian product $T_1 \times T_2$ is connected $T_1$ and $T_2$ are connected.

This is demonstrated at Product Space is Connected iff Factors 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 Cartesian product:
 * $\displaystyle \prod_{i \mathop = 1}^{k + 1} T_i = \left({\prod_{i \mathop = 1}^k T_i}\right) \times T_{k + 1}$

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