EXAMPLES OF HEREDITARILY STRONGLY INFINITE-DIMENSIONAL COMPACTA

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1 Volume 3, 1978 Pages EXAMPLES OF HEREDITARILY STRONGLY INFINITE-DIMENSIONAL COMPACTA by R. M. Schori ad Joh J. Walsh Topology Proceedigs Web: Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume EXAMPLES OF HEREDITARILY STRONGLY INFINITE-DIMENSIONAL COMPACTAl R. M. Schori ad Joh J. Walsh Examples are give of strogly ifiite dimesioal compacta where each o-degeerate subcotiuum is also strogly ifiite dimesioal. These are by far the easiest of such examples i the literature ad i additio a dimesio theoretic pheomeo is idetified which is used to verify this hereditary property. 1. Itroductio The first example of a ifiite dimesioal compactum cotaiig o -dimesioal ( ~ 1) closed subsets was give by D. W. Hederso [He] i 1967; shortly thereafter, R. H. Big [Bi] gave a simplified versio. I 1971, Zarelua [Z-l], i a relatively uow article 2 (i Russia), gives probably the simplest costructio of this type of example. Later, i 1974, Zarelua [Z-2] costructed more complicated e~amples which had the property that each o-degeerate subcotiuur was strogly ifiite dimesioal. I 1977, the authors together with L. Rubi [R-S-W] developed a abstract dimesio theoretic approach for costructig these types of examples; a sigificat feature of the latter approach was that the ey cocepts of essetial families ad cotiuum-wise separators were properly idetified. The secod author [Wa] used IThe first author was partially supported o NSF Grat MCS The authors oly became aware of [Z-l] durig the fial draft of this paper.

3 496 Schori ad Walsh this abstract approach to costruct ifiite dimesioal compacta cotaiig o -dimesioal ( > 1) subsets (closed or ot). The examples preseted i this paper have two importat features: first, their costructio is particularly simple ad clearly illustrates the pheomea uderlyig all the previous costructios; ad secod, i spite of the simplicity of their costructio, these examples have the property that every o-degeerate subcotiuum is strogly ifiite dimesioal. A pheomeo is isolated i 7 which shows that these examples are hereditarily strogly ifiite dimesioal ad ca be used to show that the "extra care" exercised i [Z-2] ad [R-S-W] i order to isure this hereditary property is ot ecessary. The secod example i this paper, see 6, uses the same costructio as i [Z-l] where rather techical proofs are used to verify the weaer coditio that the example cotais o -dimesioal ( ~ 1) closed subsets. This property follows rather automatically for us usig the theory developed i [R-S-W]. 2. Defiitios ad Basic Cocepts By a space we mea a separable metric space, by a compactum we mea a compact space, ad by a cotiuum we mea a compact coected space. We follow Hurewicz ad Wallma [H-W] for basic defiitios ad results i dimesio theory. Specifically, by the dimesio of a space X, deoted dim X, we mea either the coverig dimesio or iductive dimesio (sice these are equivalet for separable metric spaces). A space which is ot fiite dimesioal is said to be ifiite

4 TOPOLOGY PROCEEDINGS Volume dimesioal. We collect below the defiitios ad results eeded i this paper; the reader is referred to [R-S-W] for a more thorough discussio Defiitio. Let A ad B be disjoit closed subsets of a space X. A closed subset S of X is said to separate A ad B i X if X-S is the uio of two disjoit ope sets, oe cotaiig A ad the other cotaiig B. A closed subset S of X is said to cotiuum-wise separate A ad B i X provided every cotiuum i X from A to B meets S Defiitio. Let X be a space ad r be a idexig set. A family {(A,B ): E r} is essetial i X if, for each E r, (A,B ) is a pair of disjoit closed sets i X such that if S separates A ad B i X, the {S: E r} ~ ~ Theorem. [H-W, p. 35 ad p. 78]. For a space X, dim X > if ad oly if there exists a essetial family { (A ' B ): = 1,,} i X Remar. Usig the Hausdorff metric, the set of o-empty closed subsets of a compactum is a compactum. Whe we refer to a collectio of closed subsets beig dese, we mea dese with respect to the topology geerated by this metric Propositio. [R-S-W; Propositio 3.4]. Let {(A,B ): = 1,2,,} be a collectio of pairs of oempty, disjoit closed subsets of a compactum X. For each

5 498 Schori ad Walsh = 1,2,---, 3 let S be a o-empty dese set of separators of A ad B ad let Y be a closed subset of x. If for each choice of separators S E S.J = 1, 2, - - -, 3 we have that ({S: = 1,2, -- -, ) Y ~ ~.J the { (A Y,B Y) : = 1,2,---,} is a essetial family i Y ad 3 dim Y >. therefore.j 2.6. Defiitio. A space X is strogly ifiite dimesioal if there exists a deumerable essetial family {(A,B ): = 1,2,---} for X. A space X is hereditarily strogly ifiite dimesioal if each o-degeerate subcotiuum of X is strogly ifiite dimesioal Theorem. [R-S-W; Propositio 5.5]. Let X be a strogly ifiite dimesioal space with a essetial family { (A, B ): = 1, 2, }. For = 2, 3, let S be a co tiuum-wise separator of A ad B i X. If Y = {S: 2,3,---}.J the Y cotais a cotiuum meetig Al ad B l. 3. Outlie of the Example Let the Hilbert cube be deoted by Q = IT{I : = 1,2,---} where I = [0,1], let IT : Q ~ I deote the projectio, ad -1-1 let A = IT (1) ad B il (0). The family {(A,B ): = 1,2,---} is a essetial family i Q [H-W, p. 49]. For each = 1,2,---, a space Y = X X will be 3 - l 3 costructed such that: 3.1. X. cotiuum-wise separates A ad B.. J j J 3.2. If C is a closed subset of Y ad IT(C) = I, the dim C > 2; if fact, {(A 3 - l C,B C), (A C,B C)} 3 _ l 3 3

6 TOPOLOGY PROCEEDINGS Volume is essetial i C. Thus, y' = {y : = 1,2, } has the property guarateed by Theorem 2.7 that Y' cotais a cotiuum meetig A l ad B l (also A ad B ad if C is a closed subset of Y' such 3 + l 3 + l ) that for some, IT(C) = I the dim C > 2., Also a space X will be costructed such that l is satisfied as well as: 3.3. If C is a o-degeerate subcotiuum of Y" {x 3 + l : 1,2, }, the there is a iteger such that IT(C) = I The space Y = Y' Y" = {x : = 2,3, } will be a example of a hereditarily strogly ifiite dimesioal space. We will ow argue usig coditios that it is a ifiite dimesioal compactum that cotais o -dimesioal ( ~ 1) closed subsets. Theorem 2.7 guaratees that Y cotais a cotiuum meetig A l ad B ad hece dim Y ~ 1, ad l 3.2 ad 3.3 guaratee that X cotais o l-dimesioal subcotiua. The the compactess isures that X cotais o l-dimesioal closed subsets sice compact totally discoected sets are O-dimesioal. This is sufficiet sice, from the iductive defiitio of dimesio, it is clear that each closed -dimesioal ( > 1) set cotais -dimesioal closed subsets for each 0 < < ad i particular for = 1. Thus, Y is ifiite dimesioal ad cotais o -dimesioal ( ~ 1) closed subsets. I sectio 6 we prove that this example is hereditarily strogly ifiite dimesioal. 4. Costructig Y Let {Wi: i = 1,2, } be the ull sequece of ope

7 500 Schori ad Walsh l itervals i I idicated i Figure 1. Let {Si - : i 1,2,-- -} ad {S~: i = 1,2,- --} be a coutable dese sets of 1. separators of A - 3 l ad B - ad A ad B 3 l 3 3, respectively. Let a: N ~ N x N be a bijectio where N deotes the atural umbers ad let a ad a be a composed with projectio oto l 2 the first ad secod factor, respectively. Let X 3-1 = ]]~l(i - U{W i : i = 1,2, }) U (U{S~~(~) -1-1 IT (Wi): i = 1,2,---}) ad let X IT (I - U{W i 3 = i : ,2, }) U (U{ S (.) IT (W.): 1. l, 2, } ); see Figure 2 a where = 1. It is easily see that X ad x 3 cotiuum 3 - l wise separate A ad B ad A ad B respectively. 3 - l 3 - l 3 3, I additio, if C ~ x x with IT(C) = I ad (i,j) E 3 - l l 3 N x N, the C II (W 1 ) ~ s. S. ; therefore, a- (i,j) 1. J Propositio 2.5 guaratees that if C is a closed subset of X 3 - x with IT(C) = I, l the dim C > 2. 3 The ature of x 3 + l is differet tha that of X 3 - l ad X 3 : hold. the role of x 3 + l is to isure that coditio 3.3 will -1 Let x = IT,3+l(R3+l ) where IT,3+l is the projectio oto I x ad R 3 + l ~ I x is the "roof l l top" i Figure 3. x ) Fig. 1

8 TOPOLOGY PROCEEDINGS Volume Fig. 2 (!,l) Fig Verifyig Coditio 3.3 If J is a subiterval of [0,1], let i(j) deote the legth of J. Let C ~ Y be a o-degeerate subcotiuum, let i be such that IT. (C) is also o-degeerate, ad let l 1 1 i(it. (e» = E > O. Note that sice the slopes of the straight 1 1 lie segmets of R 3i + l are ±2, ad C ~ X 3i +1' the 1 ~ IT. (C) 1 1 1

9 502 Schori ad Walsh implies that ~(rr3il+l(c)) = 2 E. Iductively, let i 3i _ l +l, let I = IT (C) ad observe that if i ~ I - l, the i ~(J) = E. Sice each I has legth 1, it follows that there exists a N such that i E I N. Thus, by observig the correspodig properties of R +, it follows that 1 E 3i l I + N l that 0 E I N + = [O,b] for some 0 < b < 1. Followig the 2 above argumet we see that if! < b < 1, the I + [0,1] N 3 ad if 0 < b < i, the I = [0,2b] ad hece for some N + 3 j > 3, I N + = [0,1] which says that for some, IT(C) I. j 6. A Geeralizatio ad Let X be a strogly ifiite dimesioal compactum with essetial family {(A,B ): = 1,2, }; let {IT : = 1,2, } be a coutable dese subset of the space of all mappigs from X to I = [0,1]; for each, let {S~: 1. i = 1,2, } be a couta ble dese set of separators of A ad B ad let {Wi: i, 1,2, } be the ull sequece of ope itervals i [!,i] idicated i Figure 4. 3 o 4" 1 ), Fig. 4 X ad Let u,u,u be as before ad, for each, let Y X l 2 2 where X 2-1 = IT (I - U{W : i = 1,2, }) U i 2-1 (U {S (.) IT (W.): i = 1, 2, } ) ul 1. ' 1.

10 TOPOLOGY PROCEEDINGS Volume X 2 + l = Il (I - U{W :]. 1,2, }) U i 2+l -1 ( U{S (.) Il (W.): i = 1, 2, }). 0. ]. ]. 2 It is easily see that coditio 3.1 is true ad the earlier argumet shows that: 6.1. If C is a closed subset of Y ad Il(C) ~ [i,t], the dim C ~ 2, if fact, {(A C,B C), {A C, l B 2 + C)} is essetial i C. l Lettig Y = {y : = 1,2, } = {x : = 2,3, }, Theorem 2.7 guaratees that Y cotais a cotiuum meaig Al ad B l Sice the Il's are a dese set of mappigs the followig holds: 6.2. If C ~ Y is a o-degeerate subcotiuum of Y, the for some,il(c) ~ [i,!]. Thus our previous argumet shows that we have costructed i a arbitrary strogly ifiite dimesioal space X a subcompactum Y that is ifiite dimesioal ad cotais o -dimesioal ( > 1) closed subsets. We will show i the ext sectio that i fact Y is hereditarily strogly ifiite dimesioal. 7. Strog Ifiite Dimesioality of Subcotiua Oe reaso for the additioal complexity i the costructio i [Z-2] ad [R-S-W] was to be able to coclude that the examples had the additioal property that each o-degeerate subcotiuum was strogly ifiite dimesioal. Although we made o effort to costruct examples with this hereditary property, the followig propositios isolate a pheomeo which forces them to have this property.

11 504 Schori ad Walsh Propositio 7.1 gives coditios o a cotiuum that imply it is strogly ifiite dimesioal. Observe that coditios 3.2 ad 3.3 (resp., 6.1 ad 6.2) imply that each odegeerate subcotiuum of the example costructed i sectios 3 ad 4 (resp., sectio 6) satisfies the hypothesis of Propositio 7.1 ad thus these examples are hereditarily strogly ifiite dimesioal. A alterative argumet for the example costructed i sectio 6 ca be give usig Propositio Propositio. Let {(A,B ): :;:: 1,2, } be a famizy of pairs of disjoit czosed subsets of a cotiuum X. Suppose that~ for each ~ there are positive itegers i ad j such that~ for each cotiuum C ~ X meetig A ad B~ the pair {(A. C,B. C), (A. C,B. C) } is essetiaz i C. 1 1 J J If~ for some ~~ A~ ~ ad B ~ ~~ the X is strogzy ifiite dimesioaz. AZterateZy~ if for some i ad j~ {(Ai X, B. X), (A. X,B. X)} is essetiaz i X~ the X is strogzy 1 J J ifiite dimesioaz. Proof Let i l ad jl be such that {(A.,B. ),(A.,B. )} Jl Jl is essetial i X. Let i ad j2 be such that for each co 2 tiuum C meetig A. ad B., { (A. C, B. C), (A. C, Jl Jl J2 B. C)} is essetial i C. Recursively, for ~ 3, let i J 2 ad j be such that for each cotiuum C meetig A. ad I-l B., { (A. C,B. C), (A. C,B. C) } is essetial i I-l l 1 I I c. We ow show that the family { (A., B. ): = 1,2,. } is 1 1 essetial i X. For = 1,2,, let S separate Ai ad B.. 1

12 TOPOLOGY PROCEEDINGS Volume Sice {(A.,B. ), (A.,B. )} is essetial i X, Sl cotais a Jl Jl cotiuum from A. to B.. Sice {(A.,B. ),(A.,B. )} is Jl Jl J 2 J2 essetial i this cotiuum, Sl S2 cotais a cotiuum from A. to B. Sice { (A.,B. ), (A.,B. ) } is essetial i this J2 J J3 J3 cotiuum, cotais a cotiuum from A. to B.. Sl S2 S3 J 3 J 3 Cotiuig this argumet, for each > 1,... S co- Sl tais a cotiuum from A. to B. ad, therefore, {S: = I I 1,2, } ~ ~ Propositio. Let X be a compactum with dim X ~ 1. Suppose that, for each pair of disjoit closed sets Had K, there is a family {(A,B), (D,E)} of pairs of disjoit closed sets such that {(A C,B C), (D C,E C)} is essetial i each cotiuum C from H to K. The each o-degeerate subcotiuum of X is strogly ifiite dimesioal. Proof. Sice the hypotheses are satisfied by odegeerate subcotiua of X, it suffices to assume that X is a cotiuum ad to show that X is strogly ifiite dimesioal. Let {(Al,Bl),(Dl,E )} be a essetial family i l X. Recursively, for ~ 2, let {(A,B ),(D,E )} be such that {(A C,B C), (D C,E C)} is essetial i each cotiuum C from D - to E -. The argumet used i the proof l l of Propositio 7.1 shows that {(A,B ): = 1,2, } is es setial i X. Bibliography [Bi] R. H. Big, A hereditarily ifiite-dimesioal space, Geeral Topology ad its relatio to Moder Aalysis ad Algebra, II (Proc. Secod Prague

13 506 Schori ad Walsh Topological Sympos. 1966), Academia, Prague, 1967, [He] D. W. Hederso, A ifiite-dimesioal compactum with o positive-dimesioal compact subset, Amer. J. Math 89 (1967), [H-W] W. Hurewicz ad H. Wallma, Dimesio Theory, Priceto Uiversity Press, Priceto, N.J., [R-S-W] L. Rubi, R. Schori, ad J. Walsh, New dimesiotheory techiques for costructig ifiite-dimesioa l examp les [Wa] J. J. Walsh, Ifiite dimesioal compacta cotaiig o -dimesioal ( > 1) subsets, Bull. AIDer. Math. Soc. (to appear). [Z-l] A. V. Zare1ua, O hereditary ifiite dimesioal spaces (i Russia), Theory of Sets ad Topology (Memorial volume i hoor of Felix Hausdorff), edited by G. Asser, J. Flachsmeyer, ad W. Riow; Deutscher Ver1ay der Wisseschafte, Berli, 1972, [Z-2], Costructio of strogly ifiite-dimesioal compacta usig rigs of cotiuous fuctios, Soviet Math. Dol. (1) 15 (1974), Orego State Uiversity Corvallis, Orego ad Uiversity of Teessee Koxville, Teessee 37916