Example 3

For $n\geqslant 2$, the sphere $\mathbf{S}_n$ (TG, VI, p. 10) is simply connected. Indeed, the sphere $\mathbf{S}_n$ is the union of two closed hemispheres homeomorphic to $\mathbf{B}_n$ whose intersection is homeomorphic to $\mathbf{S}_{n-1}($cf. TG, VI, p. 12). For $n\geqslant 2$, the sphere $\mathbf{S}_{n-1}$ is connected, whence the assertion.

On the other hand, the circle $\mathbf{S}_1$ is not simply connected. Indeed, the continuous mapping $p:\mathbf{R}\rightarrow \mathbf{S}_1$ defined by $p(\theta ) =$ (cos $\theta$, sin $\theta$ ) makes $\mathbf{R}$ a connected covering of $\mathbf{S}_1$ of infinite degree, hence one which is not trivializable.