Bankston, Paul

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Marquette Universitye-Publications@MarquetteMathematics, Statistics and Computer ScienceFaculty Research and PublicationsMathematics, Statistics and Computer Science,Department of1-1-1983Coarse Topologies in Nonstandard Extensions viaSeparative ultrafiltersPaul BankstonMarquette University, paul.bankston@marquette.eduPublished Version. Illinois Journal of Mathematics, Vol. 27, No. 3 (1983): 459-466. Published Link. ©1983 University of Illinois at Urbana-Champaign. Used with permission.ILLINOIS JOURNAL OF MATHEMATICSVolume 27, Number 3, Fall 1983COARSE TOPOLOGIES IN NONSTANDARD EXTENSIONS VIASEPARATIVE ULTRAFILTERSBYPAUL BANKSTON0. IntroductionLet * be a nonstandard extension (Ol-saturated will do) of a suitablylarge ground model . If A then *A will denote the image of A in* under the canonical embedding, and *[A] will denote the set {*a aA}. If (X, -) is a topological space in then *[z] is in general no longer atopology but is a basis for what we call the coarse topology on *X. Thisis one of two natural topologies one could put on *X (the other, generatedby *z, is the "Q-topology" (see [1], [2], [4], [7], [8]) and is much finer)and is closely related to the "S-topology" (see [6], [8]) used in monadconstructions in the setting of uniform spaces.Our interest here is centered on the question of when *X (always withthe coarse topology) enjoys some of the usual separation properties. As anexample, if (R, v) denotes the real line with its usual topology then *R cannever be a To-space when * is e01-saturated. In fact, if* is an enlargement(e.g. * is Il+-saturated) then *X is never To for infinite X.As far as we know, it is an open question whether *X can be To whenX is infinite and * is Ol-saturated. However, with the help of extra settheory (notably Martin’s Axiom (MA) and the Continuum Hypothesis (CH)),we can construct extensions * in which *X can be To (even Tichonov)for a large class of spaces X.To begin with, we confine our attention to ultrapower extensions *IIo() where D is a free (that is, nonprincipal) ultrafilter on a countableset I. Then * is automatically e01-saturated (since D is countably incomplete)and its elements are equivalence classes [f] [f]o of functions f d;[30 {g"{i I" g(i) f(i)} D}.In the case of the nonstandard real line, for example, we have [f] *< [g]iff {i I "f(i) < g(i)} D. For topological spaces (X, z), if U z then*U {If] {i f(i) U} D}. (Note that X and *IX] are naturallyhomeomorphic (x *x is a homeomorphism) and that *[X] C_ *X is a densesubset. This is true for any extension *.)Received June 23, 1981.(C) 1983 by the Board of Trustees of the University of IllinoisManufactured in the United States of America459460 PAUL BANKSTONThe ultrafilters of special interest to us for the purposes of separationproperties are the so-called "separative" ultrafilters of B. Scott [10] andwill be discussed starting in 2. The reader is assumed to be conversantwith some of the more well known properties of ultrafilters on a countableset (e.g., selective ultrafilters, P-points, etc.) as well as the Rudin-Keislerorder <n/ (see [3], [5], [9]). Our set theoretic notation is standard: IAI isthe cardinality of A, OA is the set of functions f: B A, and cardinalsare initial ordinals (which are the sets of their ordinal predecessors). Thusa Ia[ for cardinals a,/3. As usual, to {0, 1, 2, ...}, and c 2.I. General Properties of *XWe will assume always that * is an tOl-saturated extension (e.g., acountably incomplete ultrapower extension) and that (X, ) is an infinitetopological space. By way of an introductory remark, it is easy to see that*It] is not in general a topology, even though it is in natural one-onecorrespondence with -. Indeed, let D be a free ultrafilter on to (in terms ofthe Stone-tech functor/3, D /3(to) \ to, where to has the discrete topology),and let-be the discrete topology on X to. Thenu {*{x} x x} *Ix]since *[X] is countably infinite and members of *It] are either finite or ofcardinal c. (If a is a countably infinite set then I*al Ino(A)l c,by a well known property of ultraproducts (see, e.g., [5]).) Thus *[-] is notclosed under arbitrary unions. It is also worthy of note that *[o’] needn"tbasically generate *[-] when o- is a basis for -. For let (X, -) be as aboveand let tr {{x} x X}. Then tr is a basis for the discrete topology;however any union of members of *[tr] will be a subset of *IX]. (Thesituation is quite different in the case of the Q-topology" *tr is always abasis for *- when cr is a basis for -.)Our first proposition is an easy consequence of the definitions involved.1.1 PROPOSITION. *X \ *IX] is nonempty and self-dense (i.e., withoutisolated points). Thus *X is never discrete.The following lemma is true for general to -saturated extensions, and isa basic result of model theory (see, e.g., [5]).1.2 LEMMA. Let (A. n < to) be a sequence of subsets of X and letm < to. If [N,<k A.I > m for each k < to then IN< *A.I > m.1.3 THEOREM. *X is a nonmetrizable Baire space which is LindelOfjustin case it is compact.COARSE TOPOLOGIES IN NONSTANDARD EXTENSIONS 461Proof. This kind of argument has been employed before (see [1], [2],[5]), so we will only sketch it here.To see that *X is Baire, let (M. n < to) be a family of dense open sub-sets of *X, and let U"be nonempty. We show *U f3 (.<First find nonempty U0"such that *U0 C_ *U M0. Using induction,we can find nonempty U.+-such that *U.+ _C *U. (f3,. M,). By(1.2), :/: n. *u. c_ n.*X is nonmetrizable: for if y *X \ *IX] and if (U. n < o) z issuch that y .< *U. then for each k < o, . U. is infinite (since*X \ *IX] is self-dense). Therefore by (1.2), [.< *U.[ > 2; so in fact,*X fails strongly to be first countable.Finally, suppose *X is Lindel6f, and let v be an open cover of *X. Wecan assume v is countable and consists of basic open sets (*U. n <Let A. X \ U.. Then n.< *A. , so by (1.2) there is a k < to withn.< *A,, t. That is, (*U. n < k) is a finite subcover of v. IA point x X is a weak-P-point if x is not in the closure of any countablesubset of X \ {x}. X is a weak-P-space if every point is a weak-P-point,i.e., if all countable subsets are closed. Clearly, a weak-P-space is T and"anticompact" (i.e., no infinite subset is compact); and a P-space whichis T is a weak-P-space. We will be concerned with these classes of spacesin the next section; for now we record the following.1.4 PROPOSITION. *X is not a weak-P-space.Proof. Let A C_ X. Then *[A] is dense in *A C_ *X. If *[A] were closedin *X then *[A] would equal *A, whence A would be finite.1.5 PROPOSITION. If X is compact then *X is compact but not T.Proof. Let (*U I) be a basic open cover of *X. Then (Ui I)is an open cover of X. If U U, is a finite subcover then *X *(UtO U U,) *U U U *U,, so *X is compact.For each x X, let/x(x) n{*u x U z} denote the "monad"of x. If *X were T then/x(x) would be {*x} for each x X; hence giveny *X \ *[X] and x X there would be a neighborhood U of x withy q *U. By compactness, then, there would be a finite subcollection ofthe U’s covering X; consequently y q *X, an absurdity. I1.6 COROLLARY. If *X is T then X is anticompact.Proof. Suppose A C_ X is compact. Then *A is compact T, whence Ais finite by (1.5).462 PAUL BANKSTON2. Separative Ultrapower ExtensionsFor the rest of the paper, we assume * Ho() for some/3(1) \ I, I countable (discrete).2.1 PROPOSITION. Suppose *X is To. Then D is "separative"" for eachpair offunctions f, g I which are distinct (rood D) (i.e., {i I f(i)g(i)} D) there is a J D such that f[J] N g[J] O.Proof. Separative ultrafilters are introduced and studied in [10]. Supposef, g I are distinct (mod D) and let h I-X be one-one. Then h f,h g are distinct (mod D). Since *X is To, it is easy to see that there is aset J D such that (h of)[J] f3 (h g)[j] . Thusf[J] g[J] O.2.2 Remark. It is straightforward to show that D fl(I) is separativeiff whenever f, g I--I are distinct (mod D) then their Stone-(ech liftingsdisagree at D (i.e., fl(f)(D) fl(g)(D)). (This condition is in fact the definitionof separativity used in [10].)We record the basic facts about separative ultrafilters which will be ofuse to us here.2.3 THEOREM (B. Scott [10]). (i) Selective ultrafilters are separative (henceMA implies the existence of separative ultrafilters).(ii) Separativity and being a P-point are not simply related.(iii) If D is separative and E <Rr D then E too is separative.(iv) If D and E are separative P-points and there is no F fl(1) \ Iwith F RK D and F RK E thenD E {R C I I" {i {j" (i,j) R} E} D}is separative (but not a P-point since D E is not minimal in the Rudin-Frolik ordering: D <RF D E).(v) D D is not separative.By (2.1, 2.3(v)) we know immediately that *X is not To whenever X isinfinite and * IIo.o(). Since there is no known proof in ZFC thatseparative ultrafilters exist, we do not know "absolutely" that coarse to-pologies can ever have any reasonable separation properties. But, giventhat D is a separative ultrafilter, quite a lot can be said in this connection.2.4 PROPOSITION.*X is Hausdorff.If D is separative and X is a Hausdorff P-space thenProof. Let [f], [g] be distinct and let J D be such that f[J] f)g[J] . Since X is a Hausdorff P-space and f[J], g[J] are countable,COARSE TOPOLOGIES IN NONSTANDARD EXTENSIONS 463there are disjoint open sets U, V C_ X with f[J] c_ U, g[J] C_ V. Thus [f]*U, [g] *V, and *U f) *V 0.Let X be any T-space and let w(X) denote the Wallman compactificationof X (see [11]). Points of w(X) are ultrafilters of closed subsets of X, andbasic open sets are of the form U {p w(X) U contains a memberof p} for U -. We identify x X with the fixed ultrafilter Px of closedsupersets of {x} and define (to) X--w(X) by(D, f) {A A C_ X is closed and f-l[A] D}(easily seen to be a member of w(X)).2.5 LEMMA. Let o oX ---> w(X) be given by o(f) 9(D, f). IfX isa weak-P-space and D is a separative ultrafilter then o induces an embeddingof *X into w(X) which leaves the points ofX fixed (i.e., o(*x) Px).Proof. Let f, g to ---> X be equal (mod D). If A 9o(f) then f-l[A]D. Now g-l[A]_f-l[A] N {n "f(n) g(n)} D, so A 9o(g). Thuso is well defined on *X. Let U -. Then [f] l[u*] iff there is closedA C_ U such that f-1[A] D iff there is a closed A C_ U such that [f]A iff [f] * U, sincef[to] is countable hence closed. Thus o is continuous.To show o[*U] o[*X] fq U, we note that o([f]) o[*U] iffthere is a closed A C_ U such that f-[A] D iff o([f]) U#, again sincecountable sets are closed.We need to show o is one-one. Suppose f, g to--X are distinct (modD) and let J D be such that 3’IJ] fq g[J] 0. Then J] o([f]) andg[J] 9o([g]), whence these ultrafilters of closed sets are also distinct.Finally, it is easy to see that points of X are fixed by o, so the proofis complete.2.6 THEOREM. Let D be a separative ultrafilter.(i) IfX is a weak-P-space then *X is T.(ii) IfX is a normal weak-P-space then *X is Tichonov.(iii) IfX is a normal P-space then *X is "strongly O-dimensional" (i.e.,disjoint zero sets are separable via clopen sets; equivalently, fl(X) is "0-dimensional" in the sense of weak inductive dimension).(iv) If X is an extremally disconnected normal weak-P-space then *Xis extremally disconnected.Proof. (i) By (2.5), *X embeds in w(X), a compact T-space.(ii) If X is normal then w(X) (X).(iii) Regular P-spaces are strongly 0-dimensional, hence their Stone-(ech compactifications are 0-dimensional. Now we can make believe thatX C_ *X C_/3(X). Thus fl(*X) fl(X), whence *X is strongly 0-dimensional.(iv) /3(X) is extremally disconnected and *X is a dense subspace.464 PAUL BANKSTON2.7 Question. Can *X ever be Lindel/)f T0?2.8 THEOREM. Let X be a normal weak-P-space such that *X is LindelOfTo. Then IxI > c,Proof. Since *X is To, D is separative. Thus we can consider X C_ *XC_ /3(X). By (1.3), *X is compact, hence equal to /3(X). Let A C_ X becountable discrete. Then A is closed in X, hence C*-embedded there (see[11]). This says that A is a countable C*-embedded subset of fl(X); whencethe closure of A in fl(X) is homeomorphic to fl(to),whose cardinality is wellknown to be 2C. Thus I’X[ > 2c, so ISl > c.2.9 Question. Is it possible for *X to be normal? Paracompact?Motivated by this question, we now turn to the special case of spaces*X where X is countable discrete (X to). First of all notice that by (2.1)and (2.6), D is separative iff *to is To iff *to is an extremally disconnectedstrongly 0-dimensional space iff/3(*to) =/3(to). (In [1] it is proved by contrastthat topological ultraproducts which are not discrete can never be extremallydisconnected unless their cardinalities exceed a measurable cardinal.) Aweak affirmative answer to (2.9) is the following.2.10 THEOREM (CH). Let D be a separative P-point (e.g., a selectiveultrafilter). Then *to \ *[to] is hereditarily paracompact.Proof. We first note that in the embedding o *to-fl(to), the imageof ,po is precisely {E fl(to) E <gr D}. (Indeed, J ,po([f]) iff f-[J]D, SO D([f]) RK D. On the other hand, if E RK D then there is somef to--to such that J E iff f-[J] D. Hence E ,Po([f]).) Thus ifD is a P-point as well as being separative then *to \ *[to] is a P-space.Now *to has an open basis of cardinality c tot, so every subset of *to\ *[to] is a P-space which is "to-Lindel6f" (i.e., every open cover has asubcover of cardinality less than or equal to to). We are done once weprove the claim (also proved in [1]): If X is an to-Lindel6f regular P-spacethen every open cover of X refines to an open partition of X. To see this,simply take an open cover v which we can assume to consist of clopensets and to have cardinality to1; say v (U(t2,< U,). Then (V : < tOl) is an open refinement of v, the members ofwhich are pairwise disjoint.We close this section with a simple observation about covering propertiesin *to for separative D.2.11 PROPOSITION. IfD is separative then *to is anticompact, and neitherto nor *to \ *[to] is LindelOf.Proof. A compact subset of *to is closed in fl(to), and infinite closedCOARSE TOPOLOGIES IN NONSTANDARD EXTENSIONS 465subsets of/3(to) are well known to have cardinality 2. Since [*to[ c, noinfinite subset can be compact.Now one of *to, *to \ *[to] is Lindel6f just in case the other is. If *towere Lindel6f, it would, by (1.3), be compact. Impossible.3. Iterated UltrapowersSuppose D, E are free ultrafilters on (countable) sets I, J respectivelyand let be given. Then, letting o) denote IIo() (to avoid confusion,we replace asterisks with the ultrafilter in brackets) we can iterate theextension process and ask whether o)e is an ultrapower extension of .The answer is well known to be "yes"; oe is naturally isomorphic (asa membership structure) to o.e. The isomorphism is defined as follows.First define z(s) __> s by (f)((i, j)) f(i)(j). One can then checkquite easily that induces an isomorphism. (o)(eM ....> (o.eM, where q--([f]o)= [q(f)lo.e.Now a natural question to ask is whether q further induces homeomorphismsbetween corresponding coarse topologies (as is the case with the Q-topology(see [1])). It is easy to see that, for U r, --[(eU] ((eU, so((eX is a continuous bijection onto (’eX. In answer to the question ofwhether ((eX is always a homeomorphism, we have the following.3.1 PROPOSITION.not an open map.Let D, E be free ultrafilters on to. Then-)e)to isProof. Since to has the discrete topology, Ce)[to] t_J,<o e){n} is openin Ce)to, hence )e)[to] is open in )e)to. Let f" to ---> Ce)to be given by f(m)e)m. Then {m "f(m) Ce)[to]} to D, so [f]o o)e)to. Now ([f]o)[g]o.e where g(m, n) m, and [h]z).e [o)e)[to]] iff{m {n h(m, n) p} E for some p} D.If o.e)j is any basic open set containing [g]o.e then J D, hence J isinfinite. Since both D and E are free ultrafilters, we can find k to to---> J such that {m {n k(m, n) p} E for some p} q D. Thus[)e)[to]], hence [)<e)[to]] is not an open set. 13.2 LEMMA.regular space.Let D, E be free ultrafilters on to. Then o)e)to is not aProof. Look at the proof of (3.1) above. If U is any basic neighborhoodof [f]o which is contained in )e)to then U must be of the form o)e)[j] forsome J D.. The closure of this set in o)e)to is easily seen to beAgain, since both D and E are free, we can find [g]o o)e)j such that466 PAUL BANKSTON{n g(n) is not constant (mod E)} D. Thus ()()[o)] is an. open set containing[fo which does not contain the closure ofany open set containing [f]o.The following shows that, under CH, ((eX and (eX can have easilydistinguishable topological types.3.3 THEOREM (CH). There are ultrafilters D, E on to such that (o.e)to isregular, but (D)(e)to is not regular.Proof. Using CH and Theorem (9.13) of [5] there are nonisomorphicselective ultrafilters D, E on to. Since both are minimal in the Rudin-Keislerordering, they satisfy the hypothesis of (2.3 (iv)). Thus D E is separative;so by (2.6), (o.e)to is a Tichonov space. However, by (3.2), (D)(e)to fails tobe even regular.3.4 Remark. Under the CH, the converses of (2.4) and (2.6 (i)) fail"there is a Hausdorff space X, not a weak-P-space, and an ultrafilter D/3(to) \ to such that )X is Hausdorff. For let D, E be as in (3.3), and letX e)to. Since ’e)to is Tichonov, hence Hausdorff, and the natural bijection(v)(e)o is continuous, we know that (Z))X is also Hausdorff. But X failsto be a weak-P-space by (1.4).REFERENCES1. P. BANKSTON, Ultraproducts in topology, General Topology and Appl., vol. 7 (1977),pp. 283-308.2. , Topological reduced products via good ultrafilters, General Topology and Appl.,vol. 10 (1979), pp. 121-137.3. D. BooTh, Ultrafilters on a countable set, Ann. Math Logic, vol. 2 (1970), pp. 1-24.4. R. W. BUTTON, A note on the Q-topology, Notre Dame J. Formal Logic, vol. 19 (1978),pp. 679-686.5. W. W. COMFORT and S. NEGREPONTIS, The theory of ultrafilters, Springer-Verlag, Berlin,1974.6. J. E. FENSTAD and A. M. NYBERG, "Standard and nonstandard methods in uniform topology"in Logic Colloquium ’69, R. O. Gandy and C. M. E. Yates, eds., North Holland,Amsterdam, 1971.7. R. A. HERRMANN, The Q-topology, Whyburn type tilers, and the cluster set map, Proc.Amer. Math. Soc., vol. 59 (1976), pp. 161-166.8. A. ROmNSON, Nonstandard Analysis, North Holland, Amsterdam, 1966.9. M. E. RuI)IN, Lectures on set theoretic topology, Regional Conference Series in Math.#23, Amer. Math. Soc., Providence, R.I., 1975.10. B. SCOTT, Points in flN-N which separate functions, Canadian J. Math., to appear.l. S. WILLARD, General topology, Addison-Wesley, New York, 1970.MARQUETTE UNIVERSITYMILWAUKEE, WISCONSIN

Coarse Topologies in Nonstandard Extensions via Separative ultrafilters

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Coarse Topologies in Nonstandard Extensions via Separative ultrafilters

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