# Topology

A Möbius strip, an object with only one surface and one edge. Such shapes are an object of study in topology.

Topology (from the Greek τόπος, “place”, and λόγος, “study”)[1] is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object.[2]

# Topology in GIS

Two adjacent polygons sharing a common border
A point-in-polygon and a polygon-in-polygon

In geodatabases, a topology is a set of rules that defines how point, line, and polygon features share coincident geometry. Topology describes the means whereby lines, borders, and points meet up, intersect, and cross. This includes how street centerlines and census blocks share common geometry, and adjacent soil polygons share their common boundaries. Another example could be how two counties that have a common boundary between them will share an edge, creating a spatial relationship.

Common terms used when referring to topology include: dimensionality, adjacency, connectivity, and containment, with all but dimensional dealing directly with the spatial relationships of features.

• Dimensionality - the distinction between point, line, area, and volume, which are said to have topological dimensions of 0, 1, 2, and 3 respectively.
• Adjacency - including the touching of land parcels, counties, and nation-states (They share a common border).
• Connectivity - including junctions between streets, roads, railroads, and rivers (Very common topological error. See diagrams about "Overshoot" below).
• Containment - when a point lies inside rather than outside an area.[3]

Topology defines and enforces data integrity rules (there should be no gaps between polygons). It supports topological relationship queries and navigation (navigating feature adjacency or connectivity), sophisticated editing tools, and allows feature construction from unstructured geometry (constructing polygons from lines).

Addressing topology is more than providing a data storage mechanism. In GIS, topology is maintained by using some of the following aspects:

1. The geodatabase includes a topological data model using an open storage format for simple features (i.e., feature classes of points, lines, and polygons), topology rules, and topologically integrated coordinates among features with shared geometry. The data model includes the ability to define the integrity rules and topological behavior of the feature classes that participate in a topology.

2. Most GIS programs include a set of tools for query, editing, validation, and error correction of topology.

3. GIS software can navigate topological relationships, work with adjacency and connectivity, and assemble features from these elements. It can identify the polygons that share a specific common edge; list the edges that connect at a certain node; navigate along connected edges from the current location; add a new line and "burn" it into the topological graph; split lines at intersections; and create resulting edges, faces, and nodes.

## Elements of a geodatabase topology

In a geodatabase, the following properties are defined for each topology:

• The name of the topology to be created.
• The cluster tolerance used in topological processing operations. The cluster tolerance is often a term used to refer to two tolerances: the x,y tolerance and the z-tolerance. The default value for the cluster tolerance is 10 times the coordinate resolution.
• List of feature classes. First you need a list of the feature classes that will participate in a topology. All must be in the same coordinate system and organized into the same feature dataset.
• The relative accuracy rank of the coordinates in each feature class. If some feature classes are more accurate than others, you will want to assign a higher coordinate rank. This will be used in topological validation and integration. Coordinates of a lower accuracy will be moved to the locations of more accurate coordinates when they fall within the cluster tolerance of one another. Features with the highest accuracy should receive a value of 1, less accurate feature classes a value of 2, even less accurate feature classes a value of 3, and so on.
• A list of topology rules for how features share geometry.
An example of overshoot before and after topological correction.

## Topological Errors in GIS

Topological errors can occur when data is scanned or digitized. They can be the result of human or computer error. They can also occur when raster data is turned into vector data. In both cases, vector lines are created but then must be further processed to form topologically correct relationships, especially when creating polygons [4]. Some examples of this include: overshoot, undershoot, sliver polygons, unclosed polygons, and mismatching adjacent polygons. Topological errors are dangerous to geographic information systems because they break the relationship between features which prohibits correct analyses from being performed[5]. These errors can be corrected by a variety of tools available in different geographic information systems. Most of these tools are able to look at problem areas and extend lines that don’t quite meet or fill in gaps in polygons within a certain tolerance that is set by the user.

## References

1. Wikipedia contributors, Topology. Wikipedia, The Free Encyclopedia. Accessed 22 June 2010.
2. Weisstein, Eric W. Topology. From MathWorld--A Wolfram Web Resource. Accessed 22 June 2010.
3. Longley et al. (2011). Geographic Information Systems and Science (3rd ed.). John Wiley & Sons, Inc: New Jersey.
4. Longley et al. (2011). Geographic Information Systems and Science (3rd ed.). John Wiley & Sons, Inc: New Jersey.