= Path-class construction of a subgroup covering
For a path-connected, locally path-connected, <semilocally simply connected> space $X$, base point $x_0$, and <subgroup> $H\le\pi_1(X,x_0)$, take <paths> starting at $x_0$ modulo the relation $\alpha\sim\beta$ when their endpoints agree and $[\alpha\beta^{-1}]\in H$. Projection sends a class to its endpoint. Over a path-connected neighbourhood whose loops die in $X$, classes $[\alpha\eta]$, with $\eta$ inside that neighbourhood, form a sheet mapping homeomorphically to it. These sheets define a <covering space>. A loop lifts closed precisely when its class belongs to $H$, so the injective induced <fundamental group> map has image $H$.
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