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

  1. $\varpi$ is surjective
  2. 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