= Solution
The based lifting criterion is
$$
\boxed{f_*\pi_1(Y,y_0)\subseteq p_*\pi_1(\widetilde X,\widetilde b)}.
$$
Under the stated <path-connected> and <locally path-connected> domain hypotheses, it is necessary and sufficient for a continuous based lift. Two based coverings are equivalent if a homeomorphism between their total spaces takes one chosen basepoint to the other and commutes with their projections.
For the usual classification by subgroups, total spaces are taken to be <connected>. They are then <path-connected> and <locally path-connected> because they cover a <locally path-connected> base. If two such coverings have the same subgroup $H$, apply the lifting criterion to each projection to obtain based maps $F:\widetilde X\to\widetilde X'$ and $G:\widetilde X'\to\widetilde X$ over $X$. Their composites and the identity maps are based lifts of the same projections. Uniqueness of lifts on a <connected> domain gives $GF=\operatorname{id}$ and $FG=\operatorname{id}$. Thus $F$ is an equivalence, proving \b[at most one <connected> based covering realizes a given subgroup].
The <connected>-total-space qualification is necessary. If arbitrary disconnected coverings are allowed, the identity cover of a circle and the disjoint union of two identity covers have the same subgroup at a basepoint in the first component, yet are inequivalent. The subgroup only records the chosen component in that setting.
The degree is the cardinality of a fibre. Over a <path-connected> base this is independent of the basepoint, by path lifting. For a <connected> total space it equals the <subgroup index> $[\pi_1(X,b):H]$, and it may be infinite.
Back to article page
