A

The neighborhoods $U$ for which the second condition holds are said to be

**covering**is a continuous map $\varpi:X'\to X$ between two topological spaces such that- $\varpi$ is surjective
- 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 $\forall i\in I$

The neighborhoods $U$ for which the second condition holds are said to be

*trivializing neighborhoods*, stressing that the restriction of $\varpi$ to the preimage of $U$ is just a lot of 'copies' of $U$ being projected on it