Topology

Topology is a set of rules and behaviors that govern how spatial features are connected to one another. The importance of topology to GIS lies in its role in data management, particularly data integrity. Here, the foundations of topology are discussed, and advances in topological research of relevance to GIS are presented.

Topology, literally, the “study of place,” is a branch of mathematics. Although topology has its roots in ancient cultures, Leonard Euler's 1736 paper on “The Seven Bridges of Konigsberg” is widely cited as the first formal study. In this work, he proved that it was not possible to take a walking path through the city of Konigsberg, Prussia (now Kaliningrad), that crossed each of the bridges in that city only once and returned to the starting point.

The subject of topology has several branches, including algebraic topology, point-set topology, and differential topology. These branches are all studied quite independently as different fields. Point-set, or general topology, is most widely used in GIS.

Topology in GIS is generally defined as the spatial relationships between connecting or adjacent features (represented as points (nodes), lines (arcs), and polygons). These topological representations are described as follows: (a) Nodes have no dimension; (b) arcs have length; and (c) polygons have area. Correct topology dictates that arcs should have a node on either end and connect to other arcs only at nodes. A polygon is defined by arcs that surround an area, and lines must have direction (e.g., upstream/downstream), left and right sides, and a start and end point (from and to).

Topology provides a way in which geographic features are linked together. For example, the topology of an arc includes its from and to nodes and its left and right polygons and can be formally described as follows:

• ARC BEGIN_NODE, END_NODE, LEFT AREA, RIGHT AREA

Formal descriptions such as this can be used to tell the computer what is inside or outside a polygon or which nodes are connected by arcs. This provides the basis for the spatial analysis and data management functions of a GIS. Spatial analysis operations require that a GIS can recognize and analyze the spatial relationships that exist within digitally stored spatial data. Topological relationships between geometric entities include adjacency (what adjoins what), containment (what encloses what), incidence (between unlike features, e.g., arcs incident on a node), and proximity (how close something is to something else).