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We show that for every subset X of a closed surface M^2 and every basepointx_0, the natural homomorphism from the fundamental group to the first shapehomotopy group, is injective. In particular, if X is a proper compact subset ofM^2, then pi_1(X,x_0) is isomorphic to a subgroup of the limit of an inversesequence of finitely generated free groups; it is therefore locally free, fullyresidually free and residually finite.

The fundamental groups of subsets of closed surfaces inject into their first shape groups