Mill, J. (Jan) van

Schrijver, A. (Lex)

English

CWI's Institutional Repository

stichting mathematisch centrum DEPARTMENT OF.PURE MATHEMATICS J. VAN MILL & A. SCHRIJVER SUPEREXTENSIONS WHICH ARE HILBERT CUBES ..., Prepub I i cation ~ MC zw 70/76 JUNE 2e boerhaavestraat 49 amsterdam PJu.nted at .the Mathe.ma.tic.al. CentJr.e, 49, 2e BoVl.haa.vu.t/uU1.t, Amt:..teJuiam. The Mathe.ma.tic.al. CentJr.e., 6ou.nded .the 11-.th o 6 Feb1u.1.cvr.y 1946, ,l.6 a. n.on-pno6,U ..i..no:ti..tu.,tlon. cumin.g at .the. pnomoUon 06 pWl.e. mathemaz.i.e.o a.nd w a.ppUc.a.tian.6. Lt .u, .6pon.60J1.e.d by .the. Ne;thvila.nd.6 Govvmme.n..t .thnough .the NetheJri.a.nd.6 Onga.n...i..za.tion 6oJt .the. Adva.nc.e.me.n.t o 6 PUJLe. Ru e.Mc..h ( Z. W. 0) , by .the Mwuc....i..pai..U:y 06 Am.6.teJLdam, by .the Un...i..ve.Jl..6Uy 06 Am.6.teJr.dam, by .the FJr.ee Un...i..veJl..6Uy a.t Am.6.t.Vl.d.am, a.n.d by ..i..ndutdn1..u. AMS(MOS) subject classification scheme (1970): 54D35, 57A20 Superextensions which are Hilbert cubes*) by J. van Mill & A. Schrijver ABSTRACT It is shown that each separable metric, not totally disconnected, topo-logical space admits a superextension homeomorphic to the Hilbert cube. Moreover, for simple spaces, such as the closed unit interval or the n-spheres S, we give easily described subbases for which the corresponding n superextension is homeomorphic to the Hilbert cube. KEY WORDS & PHRASES: superextension, HiZbert cube, Z-set, convex. *) . This paper is not for review; it is meant for publication elsewhere l • INTRODUCTION In [5], DE GROOT defined a space X to be supercompact provided that it possesses a binary closed subbase, i.e. a closed subbase S with the property that if S' c Sand ns 1 =~then there exist s0 ,s 1 ES' such that s0 n s1 = ~-Clearly, according to the lennna of ALEXANDER, every supercompact space is compact. The class of supercompact spaces contains the compact orderable spaces, compact tree-like spaces (BROUWER & SCHRIJVER [3], VAN MILL [8J)and compact metric spaces (STROK & SZYMANSKI [12]). Moreover, there are compact Hausdorff spaces which are not supercompact (BELL [2], VAN MILL [JO]). There is a connection between supercompact spaces and graphs (see e.g., DE GROOT [6], BRUIJNING [4], SCHRIJVER [11]); moreover, supercompact spaces can be characterized by means of so-called interval structures (BROUWER & SCHRIJVER [3]). Let X be a T1-space and Sa closed T1-subbase for X (a closed subbase S for Xis called T1 if for all S ES and x EX with xi S, there exists an s0 ES with x E s0 and s0 n S = ~). The superextension \S(X) of X relative the subbase S is the set of all maximal linked systems f{ c S (a sub-system of Sis called linked if every two of its members meet; a maximal linked system or mls is a linked system not properly contained in another linked system) topologized by taking { {H E \_c;CX) IS E M} IS E S} as a closed subbase. Clearly, this subbase is binary, hence AS(X) is super-compact, while moreover X can be embedded in \S(X) by the natural embedding i: X + AS(X) defined by i(x) :={SE S I x ES}. VERBEEK's monograph [12] is a good place to find the basic theorems about superextensions. In this paper we will show that for many spaces there are superextensions homeo-morphic to the Hilbert cube Q; moreover for simple spaces such as the unit interval or then-spheres S we will present easily described subbases for n which the corresponding superextension is homeomorphic to Q. Here, a classi-cal theorem of KELLER [7], which says that each infinite-dimensioml compact convex subset of the separable Hilbert space is homeomorphic to Q, is of great help. 2 2. SOME EXAMPLES In this section we will give some examples. If Xis an ordered space, then the Dedekind completion of X will be denoted by X. Roughly speaking, X can be obtained from X by filling up every gap. We define X to be that ordered space which can be obtained from X by filling up every gap with two points, except for possible endgaps, which we supply with one point. The compact space X thus obtained, clearly contains X as a dense subspace. Define G 1 = { A c X I 3 x e: X: A = ( + , x J or A = [ x,-+) } and T1 ={Ac X I A is a closed half-interval} (as usual, a half-interval is a subset Ac X such that either for all a,b e: X: if b ~ a e: A then be: A, or for all a,b e: X: if b ~ a e: A then b EA) and Ao n Al}. respectively. Notice that G1 equals T1 in case Xis compact or connected. It is easy to see that AG(X) ~ X and that AT(X) ~ X. I I What about AT (X) ? 2 Example (i) If X = I, then AG (X) = AT (X) ~I.On the other hand Ay 2 (x) h . 1 • l is omeomor~hic to the Hilbert cube Q (see section 4). (ii) If X = ~. then AG (X) '.:::'.. I and AT (X) is a non-metrizable sepa-l - 1 rable compact ordered space, which has much in common with the well-known Alexandroff double of the closed unit interval. In this case, AT2 (X) is a compact totally disconnected perfect space of weight 2~o. (The total disconnectedness of AT (X) 2 follows from the following observation: for every T0 ,T 1 e: T2 with TO n T1 = 0 there exists a TO e: T2 such that TO c To' and TO n T1 = 0 and X\T0 E T2. For every finite linked system {X\T. IT. E T2 , i dl,2, ... ,n}} it is easy to construct two i i 3 distinct mls's L0 and L1 belonging to .Q {Me:" (X) IT. <t M} • • 1.-l T2 1. (iii). showing that "T (X) is perfect. Finally "T can be embedded in 2 1 "T (X); hence weight (AT (X)) = 2~0.) 2 2 If X = R \IQ, then "G (X) :::. I, while "T (X) :::. "T (X) '.:::'.. C, the I - I - 2 -Cantor discontinuum, for it is easy to see that "T (X) and I "-y2 (X) both are totally disconnected compact metric perfect spaces. Finally define Notice that G2 equals T2 in case Xis compact or connected. Example (i) If X = I, then "G2(X) ~ Q (section 4). (ii) If X = IQ' then "G/X) ~ Q. (iii) If X = R\Q, then "G (X):::. Q. 2 -The fact that "G (Q) :::. "G ( R\Q) '.:::'.. Q can be derived from the result )..G (I) ~ Q. 2 - :12 - 2 To see this, define G' = 2 {Ac I IA e: G2 and A has rational endpoints} and " G2 = {Ac I IA e: G2 and A has irrational endpoints}-By theorem 5 and theorem 7 of [9] (cf. theorem 3.1 below), it follows that and 3. SUPEREXTENSIONS WHICH ARE HILBERT CUBES In this section we will show that for each separable metric, not totally 4 disconnected topological space X, there exists a normal closed T1-subbase S such that AS(X) is homeomorphic to the Hilbert cube Q. First we will give some preliminary definitions and recapitulate some well-known results from the literature, which are needed in the remainder of this section. A closed subset B of Q is called a Z-set ([!]), if for each s > 0 there exists a map f: Q • Q\B such that d (f, id) < s. Examples of Z-sets are compact subsets of (0,1) 00 and closed subsets of Q which project onto a point in infinitely many coordinates. In fact, Z-sets can be characterized by the property that for every Z-set B there exists an autohomeomorphism ¢ of Q which maps B onto a set which projects onto a point in infinitely many coordinates ([I]). Ob-viously, the property of being a Z-set is a topological invariant. Moreover, it is easy to show that a closed countable union of Z-sets is again a Z-set. Th~ importance of Z-sets is illustrated by the following theorem due to ANDERSON[!]. THEOREM. Any homeomorphism between two Z-sets in Q can be extended to an autohomeomorphism of Q. We will apply this theorem to show that every separable metric, not to-tally disconnected topological space X can be embedded in Qin such a way that Q has the structure of a superextension of X, i.e. every point of Q represents an mls in a suitable closed subbase for X. The canonical binary subbase for Q 1.s T = {A c Q I A = TT - 1 [ 0 , x J n -1 or A = TT [x, I], with n E :N and x E I} n and consequently, if we embed X in Qin such a way that for every two ele-ments T0 ,T 1 ET with T0 n T1 1 0 we have that T0 n T1 n X f ~. then Q is a superextension of X; this 1.s a consequence of the following theorem ([9], theorem 5). THEOREM 3.1. Let X be a subspace of the topological space Y. Then y is ho-meomorphic to a superextension of X if and only if Y possesses a binary closed subbase T such that for all T0 ,T 1 ET with T0 n T1 =f. 0 we have that To n T I n x =/- 0. 5 In particular, in theorem 3.1 Y ~ ATnX(X), where T n X = {T n X IT ET}. THEOREM 3.2. For every separable metric, not totally disconnected topological space X, there exists a normal closed r 1-subbase S such that AS(X) is homeo-morphic to the Hilbert cube Q. :N PROOF. Assume that Xis embedded 1.n Q(=I ) and let C be a non-trivial compo-of X. Choose a convergent sequence Bin C. Furthermore, define a sequence {yn}:=O in Q by for i = 1,2, ... , . It is clear that Moreover define z E Q by if if i -:/- n i = n, z. = 0 (i=l,2, ... ,). Then 1. E = {y In E :N} u {z} n is a convergent sequence and therefore is homeomorphic to B. Since Band E both are closed countable unions of Z-sets in Q, they themselves are Z-sets. Choose a homeomorphism¢: B • E and extend this homeomorphism to an auto-homeomorphism of Q. This procedure shows that we may assume that Xis embedded in Qin such a way that E c C. Let r0 ,T1 ET such that T0 n T1 # 0, where T is the canonical binary closed subbase for Q. We need only consider the fol-lowing 4 cases: -1 - - I CASE I: TO =TTnO [O,x]; T1 = TTnO [y,l] (x:2:y). Since z E T0 and y0 E T1 and C is connected, it follows that ¢-:/- T0 n T 1 n cc T0 n r 1 n x. -l -I CASE 2: TO = TTnO [O,x]; T1 = TTnl [y, I J (n0/n 1) · Then ynO E TO n T1 n X. 6 -l -1 CASE 3: T = TT [O,x]; Tl = TT [0,y]. 0 no nl Then z E TO n T1 n X. -l -1 CASE 4: T = TT [x,I]; Tl = TT [y,l]. 0 no nl Then Yo E Ton Tl n X. This completes the proof of the theorem. • 4. A SUPEREXTENSION OF THE CLOSED UNIT INTERVAL In the present section we will prove that AG 2(I) is homeomorphic to the cube, where G2 = {[x,y] lx,y EI} u {[O,x] u [y,I] lx,y EI}. For this pur-p0se we introduce F = {f:I + Ii f(O) f(x) s y - x}. 0 and if x,y EI and x s y then O ~ f(y)-Hence each f E Fis continuous and monotone non-decreasing. On F we define a topology by considering Fas a subspace of C[I,I] with the point-open to-pology. We obtain the same topology on F by ordering F partially as follows: f s g iff for each x EI: f(x) s g(x), (f,g E F), and then ta~ing as a closed subbase for F the collection of all subsets of the form {f E Flf s f 0 } or {f E Flf ~ f 0}, where f 0 runs through F. We first prove that F:;;; Q and next that AG (I)'.:::'. F; we conclude that 2 -AG (I)'.:::'. Q. 2 -THEOREM 4.1. F:;;; Q. PROOF. We show that Fis a compact, infinite-dimensional, convex subspace I of I, with countable base; hence, by KELLER's theorem, Fis homeomorphic to the Hilbert cube Q. Fis clearly a convex subspace of II; it is also clear that (F,s), as defined above, is a complete lattice, whence Fis compact. F has a countable subbase, since the collection of all subsets of the forms {f E Flf(x) ~ y} and {f E Flf(x) s y} where x,y E ~ n I, forms a countable closed subbase for F. Finally, Fis infinite-dimensional, because Q can be embedded in F. 7 For, let §! = (a 1 ,a2 ,a3 , • • .) E IE. Let G(!!!) be the smallest function f in F (in the orderings of F) such that for each i = l,2,3, .•. the following holds: It can be seen easily that G defines a topological embedding of Q 1n F. O THEOREM 4.2. AG (I)~ F. 2 -PROOF. Define a function K: AG ( I) + I by: 2 K(M) = inf{x E Ij[O,x] e: M}, and a function H: AG ( I) + F by: 2 (ME AG (I)), 2 H(M)(i) = inf{x E Il[O,xJ u [y,l] EM, x + y = K( 11 ) + 1."} (1" I M ' (I)) M ' E , E /1.G • We prove that His an homeomorphism between AG (I) and F. 2 First we observe that: K(M) S X if£ ro,x] E M; K(M) ::C: X iff [x, I ] E M; K(M) = X iff [O,x] E M and [x,I] e: M; H(M)( i) S X iff [O,x] u [K(M) + i - x, I] E M; H(M)(i) ::C: X iff [x,K(M) + i - x] EM; H(M) (i) = x iff [O,x] u [K(M) + i - x, 1 J E M and [ x,K(M) + i - x] EM; '2 these facts follow easily from the fact that M 1.s a maximal linked system in G2 . Also we have K(M) = H(M)(I). Next we show that H(M) e: F, for each maximal linked system M. In fact (i) H(M)(O) = 0, for [0,0] u [K(M),l] e: ,\ii and [O,K(M)J e: M; (ii) if is j, H(M)(i) = x, H(M) (j) = y, then x s y, for [x,K(M) + j - x] ::) [x,K(M) + i - x] EM, hence [x,K(M) + J - x] EM and y = H(M)(j):::: x; also y - x s j - i, 8 for [y - j + i, K(M) + i - (y-j+i)J::, [y,K(M) + J - y] e M, hence X = H(M)(i) ~ y - j + 1. H is a one-to-one function, for suppose M1 , M2 e AG (I), M1 i= M2 and H(M 1) = H(M2). Let a= K(M 1) = H(M 1)(1) = H(M2)(l) = K(a2), i.e. [O,a] e M1 n M2 and [a,I] e M1 n M2. Since M1 1 M2 we may suppose that there are x' and y' such that [0,x'] u [y',I] E M1 \ M2 • Since [O,a] e M2 and [a,IJ e M2 , we have x' <a< y'. Let i = x' + y' - a e [x',y'] c I. Then since [O,x'] u [a+i-x',I] = [O,x'] u [y',I] e M1 \ M2 , we find that H(M 1)(i) $ x' < H(M 2)(i) and this is a contradiction. His also a surjection. Take f e F and let: L = {[f(i), f(J) + 1 - f(i)] li EI} u {[0,f(i)J u [f(I) + l. - f(i),1]1 l. e I}. Then by definition off, it is easy to see that Lis a linked system 1.n G2• L is contained in some maximal linked system M of G2, and for this M it holds that K(M) = f(l) while for each i e I: H(M)(i) = f(i); i.e. H(M) = f. Finally we prove that His continuous. Let i,x e I. Then {ME AG (I)IH(M)(i) ~ 2 x} = ~{Me AG (I)j[O,x] u [y,l] e Mor ye 2 [O, X + y - i] EM}, and hence this set is closed. For, let Me AG (I) such that H(M)(i) this last inequality means that 2 [O,x] u [K(M) + i - x, 1 J EM. If y ?: K(H) + i - x, then [O, y + X - i]::, ro, K(M)l E M; if y :s; K(M) [O,x] u [y,l]::, [O,x] u [K(M) + i - x, l J E M. Conversely, suppose that [O,x] u [y,l] e Mor [O, x + y - i] e M, :s; x· ' + 1 - x then - x; for each ye I, then also [O, x + y - i] i M for each y < K(M) + 1 hence [O,x] u [y,I] e M; we conclude that [O,x] u [K(M) + i - x, I J E M, i.e. H(M)(i) ~ x. 9 In the same way one proves: {M E >-G(I)! H(M)(i) ;::: x} = 2 n1{M e: >-r.;(I)l[x, y] e: M ye: '2 or [x + Y - i,JJe:M}, and hence is closed. D As a consequence of these two theorems we have, as announced, THEOREM: 4.3. AG (I)~ Q. 2 5. A SUPEREXTENSION OF THE n-SPHERE In this final section we show that the superextension of then-sphere Sn with respect to the collection of all closed massive n-balls in Sn is homeomorphic with the Hilbert-cube. As usual, then-spheres Sn is the space and the closed massive n-ball with centre x E Sn and radius E;::: 0 is the set Writing B for the collection of all closed massive n-balls in Sn, we will prove that, if n;::: 1, >-B(Sn) ~ Q. Obviously ;i_ 8 (s 1) is the superextension of the circle with respect to the set of closed intervals. For the definition n+l of Bit does not matter whether the euclidian metric of R or the sphere metric of Sn (in this case the distance between~ and l in Sn is n n arccos Ei=Oxiyi' i.e. the minimum length of a curve between! and l on S) is used. However, in the proof of the theorem we need the latter metric and we call this metric d. Furthermore we define, for each point ~ = (x0 ,x 1, ••• ,xn) E Sn, the antipode~ of~ by~= (-x0 ,-x1, ... ,-xn). THEOREM 5.1. If n;::: I, ;i_ 8 (sn) is homeomorphic to the Hilbert-eu,be Q. 10 PROOF. In fact we show that AB(Sn) is compact and infinite-dimensional and n has a countable base and that AB(S) can be embedded as a convex subspace in RS0 ; hence, by KELLER's theorem, AB(Sn) is homeomorphic to Q. Clearly, AB(Sn) is compact. n To prove that AB(S) has a countable base, let X be a countable dense subset of Sn. Define B0 = {B(~,s)I! EX, £ E ~, £ ~ O}. It is not difficult to see that P: AB(Sn) + AB (Sn), such that P(M) =Mn B0 (MEAB(Sn)) is a homeomorphism; hence, sine~ AB0 (sn) has a countable base, A8(sn) also has a countable base. Next, A8 (sn) is infinite-dimensional, since AG2 (I)(:;;;:,Q) can be embedded in A8 (sn). For, let this subspace is homeomorphic to I. Let G2 be as defined in section 3, i.e. G2 is the collection of all closed subsets Y' of Y such that Y' is connected or Y \ Y'is connected. Define T: AG2(Y)--+- A8(sn) by T(M) ={BE BIB n YEM}, (MEAG2(I)). Again it is not difficult to prove that Tis a topological em-bedding. Hence AG (I):;; Q can be embedded in A8 (sn), i.e. A8 (sn) is infinite-2 dimensional. Sn Finally we embed A8 (sn) as a convex subspace in R , by means of the n sn function U: A8(s ) _. R , determined by: U(M)(~) =inf{£~ OIB(!,£) EM}, (MEAG (Sn), !ESn). 2 The mapping U is continuous and one-to-one since U(M)(!) ~ £ iff B(~,£) EM, and U(M)(!) ~ £ iff B(~,n-£) EM. And indeed, U[A8 (sn)J is a convex subspace sn n of R . In order to show this, we need only prove: if M1 ,M2 E A8(s ), then there exists an ME AB(Sn) such that U(M) = ½ U(M 1) + ½ U(M2) (U[AB(Sn)l sn being compact and hence closed in R ) . So take M1 ,M2 E A8(sn) and let M3 = {B(~,£)1~ E Sn,£ 2: ¼U(M 1)(!) +. ~U(M2)(!)}. Then M3 is a linked system, because if B(~,£) and B(y,o) E M3(!,lESn, £ ~ !U(M 1)(!) + !U(M2)(~), o ~ iu(M 1)(y) + !U(M2)(y)), then: d(!,y) s U(M 1)(!) + U(M 1)(y), and d(!,I) s U(M2)(!) + U(M2)Ci); hence d(!,Y) so+£, i.e. B(~,£) n B(y,o) f ~-Let M3 be a maximal linked system containing M3 (in fact M3 is itself a maximal linked system). Then, clearly, U(M3)(!) s ½U(M 1)(!) + ½U(M2)(!), and U(M3)(!) s ½U(M 1)(!) + ~U(M2)(!), for each x E Sn. 1 l But, since for each maximal linked system M: U(M)(!) + U(M)(~) = TI we have for each x E Sn. Thus REFERENCES [l] ANDERSON, R.D., On topological infinite deficiency, Mich. Math. J. 14(1967), 365-383. [2] BELL, M., Not all compact Hausdorff spaces are supercorrrpact. (to appear) [3] BROUWER, A.E., & A. SCHRIJVER, A characterization of supercompactness with an application to treelike spaces, Report Mathematical Centre ZW 34/74, Amsterdam, 1974. [4] BRUIJNING, J., Characterization of In and 100 using the graph theoretical representation of J. DE GROOT, in, Topological structures (ed. P.C. BAAYEN), MC tract 52, Amsterdam (1974), 38-47. [5] GROOT, J. DE, Superextensions and supercompactness, Proc. I. Intern. Syrop. on extension theory of topological structures and its ap-plications (VEB Deutscher Verlag Wiss., Berlin 1967), 89-90. [6] GROOT, J. DE, Graph representations of topological spaces, in, Topolog-ical structures (ed. P.C. BAAYEN), MC Tract 52, Amsterdam (1974), 29-37. 12 [7] KELLER, O.H., Die Homoimorphie der kompakten konvexen Mengen in Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. [8] MILL, J. VAN, A topological characterization of products of compact tree-like spaces, rapport 36, Wiskundig Seminarium Vrije Univer-siteit, Amsterdam (1975). [9] MILL, J. VAN, On supercorrrpactness and superextensions, rapport 37, Wiskundig Seminarium Vrije Universiteit, Amsterdam (1975). [lO]MILL, J. VAN, A topological property of supercorrrpact Hausdorff spaces. (to appear) [I I] SCHRIJVER, A., Graphs and supercorrrpact spaces, Report Mathematical Centre ZW 37/74, Amsterdam, 1974. [12] STROK, M. & A. SZYMANSKI, Compact metric spaces have binary bases, Fund. Math., 89 (1975), 81-91. [13]VERBEEK, A., Superextensions of topological spaces, MC tract 41 Amsterdam, 1972. 

Superextensions which are Hilbert cubes

Abstract

Similar works

Full text

Available Versions

CWI's Institutional Repository