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For questions about or involving covering spaces in algebraic topology.

Let $\pi : E \to B$ be a continuous surjective map between topological spaces $E$ and $B$. We say that $\pi$ is a covering map if for every $x \in B$, there is an open neighbourhood $U$ of $x$ such that $\pi^{-1}(U)$ is a union of disjoint open sets in $E$, each of which is mapped homeomorphically onto $U$ by $\pi$.

We call $E$ a covering space of $B$ and often refer to $B$ as the base space.

The open neighbourhoods referred to in the definition are often called evenly covered neighbourhoods.

The fibres of $\pi$ are homeomorphic, so they all have the same cardinality; this cardinality is often called the number of sheets of the covering.

Reference: Covering space.

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