Example 3.2:

The theorem can be used to prove that the sphere

$$S^n = \{(x_1, x_2, \dots, x_{n+1})\in \mathbb{R}^{n+1} / x_1^2 + x_2^2 + \dots x_{n+1}^2 = 1\}$$

is connected. Define $S^n_{\pm}$ to be the closed upper and lower hemispheres. Then $S^n_{\pm}$ are connected. The intersection of these hemispheres is $S^{n-1}$. One can now apply induction observing first that the circle $S^1$ is connected since it is the continuous image of the real line under the map

$$t \mapsto \exp(2\pi it).$$