Gap sequences on Klein surfaces

In this work we provide a possible definition for the gap sequence at a point of a compact Klein surface in an attempt to generalize the notion of Weierstrass gap sequence at a point of a compact Riemann surface. We obtain some results about the properties of these gap sequences and use them to study the Gn sets consisting of the points which have n as its first non-gap. We prove that these sets are invariant under the action of the automorphisms of the surface. We show that there are Klein surfaces of arbitrary genus such that the set G1 is non-empty (if this is the case, it is a semialgebraic subset of real dimension one). If a surface has this property, then it must be hyperelliptic. In this case, we find that the topology of the sets Gn determine the topological type of the surface.