Necessary and sufficient combinatorial conditions are established for definable homeomorphic extensions of injective maps from closed definable sets of dimension at most 1 in R^2 to the plane.
Handbook of geometry and topology of singularities
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Planar extensions in o-minimal structures
Necessary and sufficient combinatorial conditions are established for definable homeomorphic extensions of injective maps from closed definable sets of dimension at most 1 in R^2 to the plane.