© 1996, Topology Atlas

Nonstandard topology bears the same conceptual relationship to nonstandard analysis as ordinary topology does to classical analysis; namely the former provides a very fruitful generalization of the latter for the purposes of treating such basic concepts as the continuity of functions.

The word "nonstandard" actually refers to an approach to doing any kind of mathematics. This approach was devised by the logician Abraham Robinson in 1960 partly to vindicate the Leibnizian notions of "infinitesimal" and "monad," which, up to that time, had only intuitive appeal and no mathematical foundation, but mostly to create a powerful new research method for mathematics in general.

The basic tool that Robinson brought from mathematical logic is the compactness theorem. Using this, he showed how to "enlarge" ordinary mathematical structures in such a way that:

(i) certain key properties of those structures are retained by the larger ones; and

(ii) new features emerge in the enlargements, which shed light on the original structures.

For example, an enlarged real line is still an ordered field, but it makes sense to talk of the "monad" of a real number as the set of new (nonstandard) points in the enlargement that are "infinitesimally close" to that real number. This enables one to formulate the notion of continuity of a real-valued function in terms of preserving the relation of "infinitesimally close to."

If the original structure is a topological space, there are not only nonstandard points in the enlargement but nonstandard open sets as well. The monad of a standard point is the intersection of all standard open sets containing that point, and is generally infinite. A basic result, due to Robinson, is that a topological space is compact if and only if every point in an enlargement is infinitesimally close to some standard point.