We discuss the problem of extending a one-to-one correspondence between two equinumerous finite sets of points in the plane to a homeomorphism between two topological spaces containing the sets. This problem arose during the development of a computer system to merge pairs of digitized map files at the Census Bureau. This system is called conflation. Conflation requires thee fundamental steps: control point selection, triangulation, and rubber-sheeting. Pairs of points, each pair consisting of a point from each map, are selected. The selected points of one map are then used as the vertices of a specific, well-defined triangulation on that map, and for each of the triangles of this triangulation we create a triangle on the corresponding set of verticles on the other map. The set of triangles on the second map need not form a triangulation there. If they do, we show that a specific extension of the correspondence between the vertices is a homeomorphism. Moreover, the converse is also true. Also, we prove a second characterization for triangulations from which an easy detection algorithm is derived. A description of related problems follows at the end.