Embedding graphs in surfaces

The basic topological facts about closed curves in $\text{R}$² and triangulability of surfaces are used to prove the folk theorem that any surface embedding of a graph is combinatorial. A basic technical lemma for this proof (a version of what it means to apply scissors to an embedded graph) is then used to give a rigourous definition of the combinatorial boundary of a face and also to introduce a combinatorial definition of equivalence of embeddings. This latter definition is on the one hand easily seen to correspond correctly to the natural topological notion of equivalence, and on the other hand to give equivalence classes in 1-1 correspondence with the classes coming from combinatorial definitions of earlier authors.