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.