products of connected spaces are connected

Theorem 1

[1, 2]Let ${\mathrm{(}{X}_i\mathrm{)}}_{i\mathrm{\in }I}$ be a family of topological spaces.Then the product space

$$\prod _{i\in I}{X}_i$$

with the product topology is connected if and only if eachspace $X_i$ is connected.

As is true of most results in topologyinvolving products,the forward implication requires the axiom of choice.