We show that for every subset of a closed surface and every , the natural homomorphism , from the fundamental group to the first shape homotopy group, is injective. In particular, if is a proper compact subset, then is isomorphic to a subgroup of the limit of an inverse sequence of finitely generated free groups; it is therefore locally free, fully residually free and residually finite.
"The fundamental groups of subsets of closed surfaces inject into their first shape groups." Algebr. Geom. Topol. 5 (4) 1655 - 1676, 2005. https://doi.org/10.2140/agt.2005.5.1655