A question about filter-powers (and infinitesimals)

A question about filter-powers (and infinitesimals)

[Eduardo Ochs]

Hello list,

I am looking for references on the following facts about interpreting
some sentences involving infinitesimals in filter-powers instead of in
ultrapowers...

Notation:
    \I is an index set;
    \F is a filter on \I;
    \U is an ultrafilter on \I, possibly extending \F;
    if X is a topological space and x is a point of X, then
    \V_x is the filter of neighborhoods of x in X;
    Set^\F is a shorthand for the filter-power (Set^\I)/\F;
    Set^\U is a shorthand for the ultrapower (Set^\I)/\U;

Terminology:
    Set is the "standard universe";
    Set^\U is the "non-standard universe";
    Set^\F is the "semi-standard universe";
    points of Set^\I are called "sequences", or "pre-hyperpoints";
    points of Set^\F or Set^\U are called "hyperpoints".

Well, the facts. Here they are:

(1) For any (standard) function f:X->Y from a topological space to
    another, and for any standard point x in X, the following three
    statements are equivalent:

    (a) f is continuous at x;
    (b) for all choices of a triple (\I, \F, x_1), where \I is an
        index set, \F is a filter on \I, and x_1 is a hyperpoint
        infinitely close to x, then f(x_1) is infinitely close to f(x);
    (c) for the "natural infinitesimal" (\I, \F, x_1) := (X, \V_x, id),
        the hyperpoint f(x_1) is infinitely close to f(x).

(2) Any filter-infinitesimal (\I, \F, x_1) infinitely close to x
    factors through the natural infinitesimal (X, \V_x, id) in a unique
    way.

(3) We can use these ideas to lift proofs done in a certain "strictly
    calculational fragment" of the language of non-standard analysis to
    constructions done in a filter-power; and then, if we replace the
    free variables that stand for infinitesimals in our formulas by
    natural infinitesimals, we get (by (c)<=>(a) in (1)) a translation
    of our proof with infinitesimals to a standard proof, in terms of
    limits and continuity.

On the one hand, I have never seen anything published about
filter-infinitesimals, and it took me a long time to find the right
formulations for this... on the other hand, ideas similar to these
seem to be implicit in many places (see the last sections of the PDF).
However, my guess is that at least parts of (3) are new.

There are more details at:
  http://angg.twu.net/LATEX/2008filterp.pdf
  http://angg.twu.net/math-b.html

Thanks in advance for any pointers, comments, etc...
Questions welcome, of course. Cheers,
  Eduardo Ochs
  eduardoochs@gmail.com


P.S.: the diagrams in the PDF were made with:
http://angg.twu.net/dednat4.html