Proposition 5

If $n > 1$, the Euclidean sphere $S_{n-1}$ is connected and locally connected, and every point of it has an open neighbourhood homeomorphic to $\mathbf{R}^{n-1}$.

The complement of a point in $S_{n-1}$ is a connected dense set, and therefore (Chapter I, § 11, no. 1, Proposition 1) $S_{n-1}$ is connected. To see that every point has a neighbourhood homeomorphic to $\mathbf{R}^{n-1}$, we have only to project stereographically from a vertex other than the given point.

From this proposition and from Proposition 3 of no. 3 it follows that $\mathbf{R}_n^*$, being the product of two connected spaces, is connected (Chapter I, § 11, no. 4, Proposition 8; cf. § 1, Exercise 10).

The intersection of $S_{n-1}$ and a closed (resp. open) half-space determined by a diametral hyperplane of $B_n$ is called a closed (resp. open) hemisphere of $S_{n-1}$. By stereographic projection onto the diametral hyperplane, the closed (resp. open) hemisphere which does not contain the vertex of projection is mapped onto a closed (resp. open) ball of $n - 1$ dimensions, to which it is therefore homeomorphic.

If $n = 2$, we say "semicircle" instead of "hemisphere".