Topology

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Topology

In mathematics, a property is said to be topological if it survives stretching and distorting of space. Draw a circle on a rubber sheet, and no amount of stretching or distorting of the sheet will make it into a line or a point. Draw two touching circles, and no amount of stretching or distorting will pull them apart or make them overlap. It turns out that many properties of importance to spatial analysis are topological, including:

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

Connectivity: including junctions between streets, roads, railroads, and rivers

Containment: when a point lies inside rather than outside an area

Many phenomena are subject to topological constraints: for example, two counties cannot overlap, two contours cannot cross, and the boundary of an area cannot cross itself. Topology is an important concept therefore in constructing and editing spatial databases. Figure 2‑7, below, illustrates the concept of topology, using the example of two areas that share a common boundary. While stretching of the space can change the shapes of the areas, they will remain in contact (retain their adjacency) however much stretching is applied.

Figure 2‑7 Topological relationships

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