When asked if topology is a key concept of GIS, most GIS users will nod their heads in agreement. But ask these same folks about how topology is handled in shapefiles and the nodding heads give way to shrugging shoulders. In 1736, the mathematician Leonhard Euler published a paper that arguably started the branch of mathematics known as topology. More recently, the United States Census Bureau, while preparing for the 1970 census, pioneered the application of mathematical topology to maps to reduce the errors in tabulating massive amounts of census data. Today, topology in GIS is generally defined as the spatial relationships between adjacent or neighboring features.
Mathematical topology assumes that geographic features occur on a two-dimensional plane.
Because features can exist only on a plane, lines that cross are broken into separate lines that terminate at nodes representing intersections rather than simple vertices. In GIS, topology is implemented through data structure. An ArcInfo coverage is a familiar topological data structure. Adjacent lines are connected through nodes, and this information is stored in the arc-node table. Another important consequence of planar enforcement is that a map that has topology contains space-filling, nonoverlapping polygons.