Cartesian Product of Intervals is Simply Connected

Theorem
Let $n \in \N$.

For all $k \in \set {1, \ldots, n}$, let $\Bbb I_k$ be a real interval of any of the real interval types.

Let $\tau_0$ denote the subspace topology on the cartesian product $\Bbb I_1 \times \ldots \times \Bbb I_n$, induced by the Euclidean topology on $\R^n$.

Then $\struct {\Bbb I_1 \times \ldots \times \Bbb I_n, \tau_0}$ is simply connected.

Proof
The result follows from Cartesian Product of Intervals is Convex Set and Convex Set is Simply Connected.