Example 2

The numerical space of $n$ dimensions $\mathbf{R}^n$ (TG, VI, p. 1) is simply connected. It is in fact a product of simply connected and locally connected spaces (I, p. 126, prop. 4 and cor. of prop. 5 above). The same is true of every open or closed cell in $\mathbf{R}^n$. A parallelotope, an euclidean ball, open or closed, in $\mathbf{R}^n$ are simply connected because they are homeomorphic to a cell (TG, VI, p. 10, prop. 2 and I, p. 124, example).