standard approximation

A term used extensively in uses of non-standard analysis (I shall use the concepts from that set of notes, so it may be useful either to read them or be otherwise familiar with Abraham Robinson's formulation of non-standard analysis).

I describe first the concept in a metric space (say R, the real numbers, or any vector space Rⁿ with the Euclidian norm or another norm, or whatever you like), then in greater generality in a topological space. Since the concept is topological, not metric, metrics which give the same topology on the space will give the same concept of standard approximation.

• Suppose X is a (standard) metric space. Then X^* is the corresponding non-standard version of X, and X^*\X is the set of non-standard elements in the content of X.

  Suppose furthermore than x is a standard member of X. Form the formulae "d(x,y)<ε" for all standard ε>0 (if it makes you feel better to have just countably many formulae, you could just as well use the formula "d(x,y)<1/n" for all standard natural numbers n). A member (as we'll immediately see, it's always a pseudo-member) y≠x of X^* which satisfies all these formulae is called "infinitesimally close to" x. In this case, x is known as the standard approximation of y.

• Suppose X is a (standard) topological space which is a Hausdorff space. Again, X^* is the corresponding non-standard version of X.

  If x is a standard member of X, we form the formulae "y∈U_x" for all neighborhoods U_x of x. A member y≠x of X^* which satisfies all these formulae is called "infinitesimally close to" x. Again, y will of necessity be non-standard, and x is called the standard approximation of y.

Note that any standard x is its own standard approximation.

Uniqueness

There is at most one standard approximation to any y in X^*. In the metric case, we note that if x and z are both standard approximations to y, then d(x,y)<1/n and also d(z,y)<1/n for all n. So from the axioms for the metrics we have d(x,z)≤d(x,y)+d(z,y)<2/n. But x,z are standard, and therefore d(x,z) must also be standard; it follows that d(x,z)=0, and x=z.

In the topological (but Hausdorff) case, suppose y∈U_x and y∈V_z for all neighborhoods U_x of x and V_z of z. If x≠z, take standard disjoint neighborhoods U of x and V of z (they exist, by the Hausdorff property of X) and apply them. Then y is a pseudo-element of U∩V, but also the formula "U∩V=∅" is true in the standard world, therefore true in the non-standard world too, and therefore U^* and V^* can have no common elements. So we must have x=z.

Existence

If X is finite, then it has no non-standard pseudo-elements, and standard approximation is not a useful property.

If X is infinite, then some of its elements may be standard approximations of non-standard pseudo-elements. But a non-standard element of X^* need not, in general, have a standard approximation. Indeed, it may happen that no non-standard element of X^* has a standard approximation. For instance, if X=N (the natural numbers) and y is a non-standard pseudo-element of X, y has no standard approximation: for any standard element x of N, it is true that "∀y∈N.d(x,y)<1 → x=y". In particular, if x were the standard approximation of y, then certainly "d(x,y)<1" would hold, and therefore also "x=y", meaning y=x is standard.

If X=R (the real numbers), then some non-standard pseudo-elements of R have a standard approximation and some do not. Let r be the set of non-standard elements whose standard approximation is 0. Then for any standard real x, the set of non-standard elements whose standard approximation is x is r_x=x+r. Proving this is a technical exercise.