Fenstad, Jens Erik

Nyberg, Anders M.

NORA - Norwegian Open Research Archives

Matema.tisk seminar Universitetet i Oslo STANDARD AND NON-STAND_/illD METHODS IN UNIFORM TOPOLOGY by J.E. Fenstad and A.M. Nyberg No 6 - 1969 - 1 -STANDARD AND NON-STANDARD METHODS IN UNIFORM TOPOLOGY. By J.E. Fenstad and A.M. Nyberg. The purpose of this note is to discuss the relationship between standard and non-standard concepts in uniform topology. W8 have~ in particular 9 been interested in the case where the space carries both an algebraic and uniform structure 9 e.g. as in the case of a topological group. We assume that the reader is familiar with the standard theory as presented e.g. in [3] and the non-standard theory as presented in [6]. Recently two contributions to the non-standard approach to uniform topology have been published ([4] 9 [5]). There seems to be little overlap with the present discussion. We believe that our emphasize on the notion of a E2unde1 point leads to a very clear understanding of the exact relationship between stan-dard and non-standard concepts. BOUNDED POINTS Let (X 9 1L) be a uniform space and let *X be a non-stan-dard extension of X • By the s-tQ£OloEY on *X we understand the topology defined by the neighbourhood systems Nx = [*U(x) I U ~ ~l 9 x E *X . - 2 -Remark. The uniformity ~~ can also be defined by an associated family of pseudo-metrics 9 jJ . And it is easily shown that the s-topology is generated by the following family of open sets in *X 9 where Sd(x,r) = [y The monad, IJ (x) , of a llOint x E .,x-x is defined to be the set Remark. Using the associated family of pseudo-metrics we see that u(x) = [y c *X I d(x,y) 0 Q. for all d t ~J } • One easily notes that the relation x c u(y) is an eq_uiva-lence relation on *X which we denote by x ~ y . The space X is Hausdorff iff every monad contains at most one standard point. We call x c *X near-standard if x belongs to the monad of a standard point. In the seq_uel we assume that X is Hausdorff in the asso-ciated topology. DEFINITION. The set Bx of pounded~oints of *X is defined to be the closure of X in *X with respect to the s-topology, - 3 -We shall obtain a characterization of B in terms of Cauchy z-ultrafilters on X . It is known (see e.g. [2] that to every x E *X there is c-associated a unique z-ultrafilter tfx on X and to every z-ultrafilter JC on X there corresponds a point x ~ *X such that Y = c.?: . ( g- is the unique z-ul trafil ter which X X extends the prime z-filter c;;--' - r n ~ -,r l x E *F cr x - L 1! ~ 1~ i and F E Z (X)}. ) PROPOSITION. The set of bounded~oints consists exactly of those x ;: ->tX such that the associated z-ul trafil ter is Cauc4:z.~ i.e. Bx = [x c *X ~ is a Cauchy z-ultrafilter1 . We sketch the proof. Let x .,: cls X ~ we have to show that ~ is Cauchy. Pick any U E 1J~ and choose a closed V E 1t such that V o V = U and such that *V(x) n X I ¢ . Let p c * V ( x) n X vV e now observe that V ( p ) E J[ ~ since X x ;.;:: -)'"V(p) and we may assume that V(p) E Z(X) • And obviously V(p) x V(p) ~ U 9 which shows that ~ is Cauchy. Conversely 9 let ~ c:r_ be Cauchy. X Let U E 1t and pick a (symmetric) u1 c 1£ such that u1 0 u1 is Cauchy~ there is a V E ~ such that c U Since now possible to pick an x 0 E ~...l (x) such that x c *V . 0 shows that (p,x0 ) E *U1 9 hence p c *U(x) n X . It is This Remark. We mention here the following result : X is com~~ete iff every bounded point is near-standard, This generalizes the fact that compactness is equivalent to every (non-standard) point being near-standard. - 4 -THE CONITv'rUTATIVE DIAGRAM. We noted above that if x E *X then 'J.(x) contains at most one standard point. If p is a standard point in u(x) then p is uniquely determined. We call p the standard part of x and write p = st(x) • Let yX denote the completion of X • Then X is imbed-ded in both *X and yX and there is a surjection n: Bx ~ yX such that the following diagram is commutative: The definition of n is immediate: TI(x) , x E B is the equivalence class of the Cauchy z-ultrafilter G:""' or in X yX • PROPOSITION. Let (X,1l) and (Y~~) be uniform spaces, and assume that (Y, ?.)-) is COJ!Qlete. Let f: X ...... Y be a uniform~ 2ontinuous map, hence f has an extension to a continuous map 1\ f: yX ...... Y . The following ident~~is valid for all x E Bx : 1\ st(·*f(x)) = f(n(x)) . As a preliminary remark toward a sketch of the proof, we note that if f: X ...... Y is uniformly continuous, then f(u(x)) c u(f(x)) for all x -: *X • This follows immediately from the definition of uniform continuity~ and, in fact, charac-terizes this notion. Let be given. Since g;-cr.. is X Cauchy and y is - 5 -complete 9 we see that the z-filter f"'( J?;c) = [Z E Z(Y) I f- 1 (Z) E ~} converges toward the point 1\ f ( TI ( X ) ) in Y Pick a point X 0 such that x c 1<-F 9 for 0 all F c $ X Then Z E f ·~ Jf . Since X ;: 'l(X) 0 and 1<-f(x ) E >*Z 0 for all f'~( ~) 1\ converges to f(n(x)) , we see - ll?<-V ( ~ ( :r ( X ) ) ) = 1\ \..t(f(n(x))). VE ?J' that The uniform continuity of f now implies that ''tf(x) 1\ E u(*f(x )). 0 Hence 7(-f(x) c 'J.(f( n(x))) 9 vvhich exactly means that 1\ st(*f(x)) = f(n(x)) . Remark. This generalizes a result in [2] where we considered v the relationship between -;:-x and the Store-Cech compactifi-cation 9X of X . In the first section of this note we showed that the set Bx 9 vvhich was defined as the s-closure of X in *X 9 is the set of points x such that (?-' J?x is a Cauchy z-ultrafilter on X . From the observations of this section it follows that 1\ st d(x,y) = d(n(x) 9 "T(y)) 9 for all bounded x and y and all d in the associated family of pseudo-metrics. This identity 11 explains 11 why the s-topology as defined by A. Robinson is the appropriate setting for dis-cussing the completion of metric spaces. It also implies that x y (i.e. x E u(y)) iff n(x) = n(y) These observations taken together shows that we obtain the completion of X as the set of bounded points modulo monads. Implicit in our observations is also the fact that this non-- 6 -standard approach is nothing but the "lifting 11 from !3X to -~,"X of a well-known procedure (see e.g. [3]). But something may be gained. A. Robinson [7] constructed R as Qf/Qi , where Qf is nothing but the bounded points of Q and Q. are the infinitesimals. l The important point here is that the algebraic operations of Q extends to Qf 9 i.e. Qf is a "nice 11 substructure of +."Q vrhich itself is an elemen-tary extension of Q . We will return to this point in the next section. :CXTBN:OING MAPS FROM X TO yX • Let as above (X 9 1l) and (Y 9 ~~) be uniform spaces, and assume that (Y, '/.7) is complete. It is known in the metric case (and fairly straight forward to extend to the uniform case) that a map f~ X ~ Y is conti~us, iff f is s-continuous for all standard x 9 or, equivalently 9 iff f(!J.(X)) ~ !J.(f(x)) for all standard x • Here 11 standard it can be replaced by 11 near-standard 11 • And a map f ~ X ~ Y is ,:unifo_:t:_'J:LlJ-J:___£,_Sll1j:;inuous 9 iff f is s-con tinuous for all x c -x-x 9 or, equi val en tly, iff f(u(x)) ~ ~(f(x)) for all x ~ *X . In this section we characterize in a similar way the pro-·· perty of having a continuous extension from X to yX • PROPOSITION. Let (X, 71) and (Y,?J) be uniform spaces 9 and assume that is com..:e._lete. Let f be a map from X to Y . The following. three_ <?~mdi tion_s. are eg_uivalent. (i) f has a continuous exten_::Ji~I}--~ yX • - 7 -(ii) f is s-continuous for all,bgund~d x c *X . (iii) f(~(x)) c u(f(x)) for all bounded x t *X . 1\ We first prove that (i) implies (ii) & (iii). Let f be the continuous extension of f to Ve consider X as a denAe subspace of yX and we vrork with non-standard extensions 1\ The map f is continuous from yX to Y • Hence /\ ''"f" maps near-standard points of ->r·yX to near-standard points in o,<-y • And -~-~(~(x)) c !-l( ·>~~(x)) , for all near-standard (Here 1\ u denotes the monad in yX.) First 9 note that near-standard is the same as bounded in *yX since yX is complete. Next, note that X is uniformly imbedded as a dense subset of yX • This means 9 in particular 9 that ~1(x) , the monad of a point x E *X 9 is the restriction of the monad 1\ 11 (x) to (This follows from the correspondence between entourages in X and yX.) Further, a point x ::: -x-x which is bounded in -K-X remains bounded in *yX . Putting these things together we see that (i) implies (ii) and (iii). As an example we verify (iii). Thus let x be a bounded point in -)E-x Hence is bounded, and therefore 1\ near-standard in yX The continuity of f then implies that 1\ 1\ 1\ 1\ -;<-f(:J.(x)) 5::: u(f(x)) . Let y c !..t(x) • Then y c !..t(x) 9 hence 1\ 1\ 1\ "'"f(y) ··~ !.l(-*f(x)) • But ¥f = -),Cf -:;-x, hence *f(y) E u(*f(x)L which vvas to be proved. 1\ Remark. Since f maps near-standard points of *yX to near--standard points in *Y it follows that f maps bounded points in -~ex to bounded points in -:t-y - 8 -The s-continuity of f on By implies immediately that .L~ f(u.(x)) c u(f(x)) for any X - Bx . In fact~ let X be boun-ded and consider an s-open neighbourhood v~ of f(x) in 'l!.-y . It then exists an s-open neighbourhood v of X such that f(V n Bx) = V' . As •J.(x) is the intersection of all s-neigh-bourhoods of x and u(x) c Bx ~ we see that f(u(x)) - V' • Since V' is arbitrary 9 the result follows. For the final part of the proof assume that f(u(x)):::(f(x)) CJ: for all bounded x . Let ~ be an arbitrary Cauchy z-ultra-filter on X . We have to show that ,, r;-' f ,Y.. = [ Z ~ Z ( Y) f- 1(z) ~ JC} is a Cauchy z-filter on Y. Note first that for some bounded point x . Consider the family f( 7;_) = [f(F) \ F ·~ ~} . Tt suffices to show that for all v :_ rzr there is some such that F' x F' = V . We prove this by contradiction. Assume not: Then there 0,w · exists a V E u such that the following (standard) sentence is true: r;--(\-/F) (F E Ji' _, (f >(f) (F) ¢ V) • A familiar type non-standard argument now gives an internal ,-;-' set F E *cf such that F ·= '.l(x) and such that 0 0 --(f xf)(F0 ) ¢ *V . Pick a symmetric W .:. 2}' such that W o W = V It follows that ( f x f) ( F ) ¢ ~(-w o ~(-w • 0 And since F c: '.l(x) 0 we further obtain that (f x f) (t.L(x)) '/- '*-W 0 -,'(-w • We may then choose points U 9 V (; 'l(x) such that (f(u) 9 f(v)) f. *W o *W 9 which easily implies that f(u(x)) ¢ -;;,W(f(x)) . However 9 - 9 -u(f(x)) c ~-W(f(x)) ~and the result follows. Remark. We note the following supplementary characterization of the set of bounded points~ A point x is bounded~ iff for all uniformly continuous f~ X ~ R ~ f(x) is bounded 5 or 5 equivalently~ iff for all f~ X - R which has a continuous ex-tension to yX , f(x) is bounded. It remains to show that if x ~ *X - Bx ~ then there is some uniformly continuous f~ X ~ R such that f(x) is not counded. Note that X can be imbedded into a product RI where I = [f~ X -• R l f is uniformly continuous1 . If x f. B , then c;z-c.Tx is a z-ul trafil ter which is not Cauchy. Hence there must be some f -c: I such that prf( ~) is a base for a z-ul-trafilter which is not Cauchy. But this means that the z-ultra-filter ~·(x) cannot be Cauchy, hence f(x) is not bounded. Let X now carry both an algebraic and uniform structure 9 e.g. let X be a topological group. When is yX an algebraic structure of the same kind as X ? The answer must be related to the "degree of continuity 11 of the algebraic operations. Con-tinuity is known to be too weak and uniform continuity too strong. It turns out thnt s-continuity of the algebraic opera-tions on the set of bounded points of ~:-x is the right kind of requirement. For simplicity assume that as an algebraic structure has certain operations f 1 , ••• ,fk and that the axioms which X is supposed to satisfy are open, positive sentences (e.g. let X be a group with both group multiplication and inverse opera-t · ) 1'T t · d fr ~ yn ~ y X . lon . v\ e mus now consl er marJs . ~~ (Note that - 10 -monads and bounded points con~ute with finite cartesian products 9 hence our previous results apply.) The s-continui ty of the or)e-rations fr implies that each fr maps bounded points to bo~m­ded points. Hence tbe operations fr 9 which can be extended to -)~x by general model-theoretic considerations 9 can be re-stricted to the bounded points. Since s-continuity means that f ('_t(b 1 ) 9 ... ,u(b )) = u(f(b 19 , •• 9 b )) 9 the operations fr can r n - n be further defined on yX ~ B/u . And since yX is a homomorphic image of a subsystem of *X 9 which itself is an elementary extension of X 9 the syn-tactic form of the axioms implies that they are also valid in yX • Thus in this case the s-continuity of the algebraic ope-rations ensures that yX is an algebraic structure of the same kind as X 9 and that the extended operations are continuous in the associated topology. And in this case the condition of s-continuity is also necessary. This includes known results on topological groups and rings. When the axioms are of a more complicated syntactic character 9 the situation becomes more involved. It is perhaps somewhat doubtful whether a useful general theorem can be stated. - 11 -REFERENCnS~ [1J. J.E. Fenstad, A note on_ 11 stand~ersus "non-standardn topology, Indag. Math. 29(1967) 9 pp 378-380. [2]. J.E. Fenstad 9 Non-stan~~r~ ~odels for arithmetic and ana-lxsis, Proceedings of the XV Scandinavian Mathematics Congress, Springer Verlag (Forthcoming). [3]. L. Gillman and M. Jerison, Ring§ of Continuous functions. Van Nostrand, Princeton (1960). [4]. W.A.J. Luxemburg, A~neraJ-~_the_or_y of monads, in W.A.J. Luxemburg (editor), Ap~lications of Model Theory to Algebra. Analy~is~ and Probability, Holt, Rinehart and Winston, New York (1969), pp 18-86. [5]. M. Machover and J. Hirschfeld, I1ectures on Non-Standard Analysis, Springer Lecture Notes, Springer Verlag, Berlin, Heidelberg (1969). [6]. A. Robinson, Non-standard analYsis, North-Holland Publ. Comp., Amsterdam (1966). [7]. A. Robinson, Non-standard ar~~thme!ic!!_, Bull.Am.Math.Soc. 9 73(1967), pp 818-843. 

STANDARD AND NON-STANDARD METHODS IN UNIFORM TOPOLOGY

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