Cartesian Product of Intervals is Simply Connected
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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.
$\blacksquare$