Examples 15.1:

We now present four examples to illustrate the concept of a covering projection.

1. As indicated in the beginning the most basic example is the map ex$: \mathbb{R} \longrightarrow S^1$ given by
      ex$$(t) = \exp(2\pi i t)$$
   For each point $z$ on the circle take an arc $U$ centered at $z$ and of length say $\pi/2$. The reader may check that the inverse image of $U$ under ex is a disjoint union of open intervals on the line.
2. Consider the map $T: \mathbb{C} - \{0\} \longrightarrow \mathbb{C} -\{0\}$ given by $T(z) = z^2$. If we pick a point $z\in \mathbb{C} -\{0\}$ and a small disc $U$ centered at $z$ not containing a pair of antipodal points then $T^{-1}(U)$ is a disjoint union of two open sets each of which is mapped bijectively onto $U$ by $T$.
3. Consider the map $p:\mathbb{C} - \{\pm 1, \pm 2\} \longrightarrow \mathbb{C} - \{\pm 2\}$ given by
   $$p(z) = z^3 - 3z$$
   The equation $p(z) = w$ has three distinct roots for each $w \in \mathbb{C} - \{\pm 2\}$ and the roots are continuous functions of $w$. For a sufficiently small neighborhood $U$ of $w$, $p^{-1}(U)$ is a disjoint union of three open sets each of which is mapped onto $U$ homeomorphically onto $U$ by the open mapping theorem. Several examples of this type related to complex analysis are discussed in [6].
4. Consider the quotient map $\eta: S^n \longrightarrow \mathbb{R}P^n$. We show that $\eta$ is a covering projection. Let $U_1$ be an open subset of $S^n$ not containing a pair of anti-podal points and
   $$U_2 = \{-x/ x \in U_1\}.$$
   Then, $\eta(U_1) = \eta(U_2)$. Denoting these images by $U$, we see that $\eta^{-1}(U) = U_1\cup U_2$ which is an open set in $S^n$ and so $U$ is open in $\mathbb{R}P^n$. Second, $\eta$ maps each of $U_1$ and $U_2$ bijectively onto $U$. To see that $\eta$ maps each of $U_1$ and $U_2$ homeomorphically onto $U$, we merely have to show that $\eta$ is an open mapping. So let $V_1$ be an open subset of $U_1$ and $V_2 = \{-x/ x \in V_1\}$. Then
   $$\eta^{-1}(\eta(V_1)) = V_1\cup V_2$$
   is open in $S^n$ so that $\eta(V_1)$ is an open subset of $\mathbb{R}P^n$. Thus we have shown that $\eta$ restricted to each $U_j$ is an open mapping and that suffices for a proof.

We now summarize the most basic properties of covering projections.