A **covering** is a continuous map $\varpi:X'\to X$ between two topological spaces such that

```
<ol>
<li>$\varpi$ is surjective</li>
<li>for all $p\in X$, there exists a neighborhood $U$ such that $\varpi^{-1}(U)=\bigsqcup_{i\in I} V_i$, with $V_i\subset X'$ open and $\varpi|_{V_i}: V_i \to U$ homeomorphism $\forall i\in I$</li>
</ol><br><br>
```

```
The neighborhoods $U$ for which the second condition holds are said to be <em>trivializing neighborhoods</em>, stressing that the restriction of $\varpi$ to the preimage of $U$ is just a lot of 'copies' of $U$ being projected on it<br><br>
```