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