Topology

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Topology

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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 however much stretching is applied.

Figure 2‑7 Topological relationships

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Topological properties are those that cannot be destroyed by stretching or distorting the space. In this case the adjacency of the two areas is preserved in all the illustrations, no matter how much distortion is applied