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A covering sheet over an evenly covered neighbourhood $U$ is one of the disjoint open subsets of $p^{-1}(U)$ mapped onto $U$ by a homeomorphism obtained by restricting the covering map.

For a covering map with nonempty connected base, the degree of a covering map is the common cardinality of its fibres. Each evenly covered neighbourhood makes this number locally constant, so connectedness makes it constant. The total covering space need not be connected. On a path-connected base the path lifting theorem gives a fibre bijection by path lifting. A disconnected base may have different degrees on different connected components.

An open neighbourhood $U$ of a point in the base of a covering map $p$ is an evenly covered neighbourhood if $p^{-1}(U)$ is a disjoint union of open subsets, each mapped onto all of $U$ by a homeomorphism obtained by restricting $p$.

For a covering map over a nonempty connected space, the degree of a covering map isthe cardinality of any fibre of a function. To show independence of $x$, choose an evenly covered neighbourhood $U$. Each covering sheet contains exactly one point over every $y\in U$, so $|p^{-1}(y)|$ equals the number of covering sheets over $U$. The fibre of a function cardinality is therefore locally constant. For a fixed cardinal number $\kappa$, the points with that cardinality form an open subset, and their complement is also open. Connectedness forces the value to be constant everywhere.

For a path-connected base there is also an explicit fibre bijection by path lifting: lift a path from $x$ to $y$ starting at each point over $x$ and take its endpoint. The path lifting theorem gives existence and uniqueness; path reversal gives the inverse bijection. The bijection may depend on the path, but the resulting equality of cardinalities does not.

A connected base is needed for one global degree. Without it, use a degree on each connected component on which the cardinality is constant. For example, a covering map of a discrete two-point topological space can have one point over the first point and two over the second. Thus the unqualified printed definition needs the usual connected-base convention, or the explicit assumption that every fibre of a function has the same cardinality.

A covering map is a continuous surjection $p:\tilde{X}\to X$ such that every $x\in X$ has an open neighbourhood $U$ for whichwhere the $U_{\alpha }$ are pairwise disjoint open subsets of $\tilde{X}$. Such a $U$ is an evenly covered neighbourhood, and the sets $U_{\alpha }$ are its covering sheets. The index set may be infinite. The requirement that every sheet map onto all of $U$ is part of the definition.