Definition 15.1:

A covering projection is a triple $(\tilde{X}, X, p)$ where $\tilde X$, $X$ are connected topological spaces and a continuous map $p: \tilde{X} \longrightarrow X$ satisfying the following properties:

(i)

The map $p$ is surjective

(ii)

Each $x \in X$ has a neighborhood $U$ such that the inverse image $p^{-1}(U)$ is a disjoint union of a collection open subsets $\{U_{\alpha}\}$ of $\tilde{X}$.

(iii)

Each $U_{\alpha}$ is mapped onto $U$ homeomorphically by $p$.

The neighborhood $U$ described in the definition above is called an evenly covered neighborhood of $x$ and the open sets $U_{\alpha}$ are referred to as sheets lying above $U$. This terminology will be used frequently. We shall also say that $\tilde X$ is a covering space of $X$ when it is fairly clear what the map $p$ is. in