A covering over a disconnected base can have different sheet numbers on different components

Given: The objects, hypotheses, and choice principles stated above.

For a covering $p:E\to B$, the cardinality of $p^{-1}(b)$ is locally constant as a function of $b\in B$. If $B$ is connected, all fibres are equinumerous. (The cardinality of a covering fibre is locally constant and is constant on a connected base).

Throughout, a topology is as in def-topological-space, and finite, at most countable and uncountable are as in def-countable, so that "countable" always means "at most countable" and every finite set is countable. Let $X$ be a set. The six families below are topologies on $X$; that each really satisfies (T1), (T2) and (T3) is discharged in full after the list. Among those six is the discrete topology ${\mathcal{T}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}:=\mathcal{P}(X)$, in which every subset is open and hence every subset is also closed. (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).