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Topology (from Greek topos, ‘place’), in mathematics, began as a generalization of the results of analysis. Part of this generalization is the use of metrics, which take the idea of distance and axiomatize it. Topological spaces are one step further, and lose the idea of distance totally while still making it possible to show some of the most useful results of analysis. Topology takes the basic notion to be that of the ‘open set’, which can be defined using distances (it is a set in which around every point, all the other points within a certain distance are also within the set), and axiomatizes the relationships between collections of open sets. (This means that anything can be called an open set provided the collection of ‘anythings’ satisfies the conditions laid down for relationships between open sets.) The conditions are that the empty set and the whole set of points are both open sets, that the set of points in both of a pair of open sets is also an open set, and that the set of points in at least one of the members of any collection of open sets is also an open set. Any system of open sets which satisfy these axioms is called a topological space.
The reason that many of the results of analysis are true of topological spaces in general is that the concept of continuity can be easily defined for topological spaces. A continuous function is one where given an open set X, the set of points mapped by the function to points in X is also an open set. (So it preserves the open set structure in a sense.)
Topology is a subject which has many applications, reflecting its generality. Another way of looking at it (or, more strictly, at one particular topological space, that of three-dimensional space) is that a continuous function is one which stretches and bends spatial objects without tearing them. For example, a mug and a doughnut are the same as far as topology is concerned, because it is possible to continuously deform one to give the other; a ball and a doughnut are not, because it is impossible to get rid of the hole in the doughnut without tearing it.
One of the most famous questions in topology, the ‘four colour theorem’, about the ways to colour regions on a map, has recently been solved, but its solution has led to controversy about the nature of proof (see provability).
Another famous result in topology is the ‘hairy-ball theorem’, which states (when put into non-technical language) that it is impossible to comb the hair on a ball that is covered with it without leaving tufts or ridges. (This result does also have serious consequences.) SMcL |
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