IMBEDDING COMPACTA INTO CONTINUA

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1 Volume 1, 1976 Pages IMBEDDING COMPACTA INTO CONTINUA by Michael Laidacker 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 IMBEDDING COMPACTA INTO CONTINUAl Michael Laidaeker As we shall show i this paper, a compactum ca be imbedded i a cotiuum i such a maer that certai properties of the compoets of the compactum are shared by the cotiuum cotaiig the compactum. A compactum is a compact metric space ad a cotiuum is a coected compactum. 1. Prelimiaries Bellamy (see [1] ad [2]) defies a pseudocoe to be a Hausdorff compactificatio S of a half ope iterval [a,b). Lettig i: [a,b) ~ of the pseudocoe. S be the ijectio map, i(a) is the vertex I additio, if X is homeomorphic to S\i[a,b), the S is a pseudocoe over X. For ffi a collectio of sets, we defie I ffi I = UC E ffi C. A. Lelek.. The followig lemma was poited out to the author by 1.1. Lemma. Let X be a compactum ad let C be the usual Cator set. The X ca be imbedded i C x X by a fuctio h such that {c} x X cotais at most oe compoet of h(x) for c E c. Refer to [15], page 148. Now suppose X is a compactum. The lettig Q be the Hilbert cube ad C be the Cator set, we ca assume without loss of, geerality that XC C x Q i such a maer that {c} x Q cotais at most oe compoet of X for c E C. Defie Co to be the subset of C such that c E Co if ad oly if ({c} x Q) X ~ ~. I additio, defie D to be the set of all compoets of (M x Q)\(C x Q) where M = [-1,2]. o IThe author ackowledges his gratitude to H. Cook ad A. Lelek for may valuable suggestios.

3 92 Laidacker If x,y E M x Q, the x is to the left of y if ad oly if the first coordiate of x is less tha the first coordiate of y. Also if K ad K' are two coected spaces i M x Q, the K is to the left of K' if x E K ad y E K' implies x is to the left of y. The term "right of" is defied i a aalogous maer. Defie D ad D 2 to be the two elemets of D such that D l l is the left most compoet of (M x Q)\(C x Q) ad D is the o 2 right most compoet of (M x Q)\(C x Q). It might be the case o that D cosists of oly two elemets. This case occurs whe X cosists of oe compoet. We will essetially be cocered with the case where D cotais a ifiite umber of elemets. For D 3, D 4,, we let K(L,i) be the left compoet of D X i for i = 3,4,.. Similarly, the right compoet of D i X is deoted K(R,i). Last of all, defie e = 4/, = 1,2,., U(K,e ) = {B(x,e ): x E K}, ad IU(K,e ) I = U KB(x,e). XE 2. Costructio of I (X) I this sectio we costruct a imbeddig of a compactum X ito a cotiuum I(X) by adjoiig a coutable umber of ope itervals ad two half ope itervals i a prescribed maer so as to ivariable preserve certai properties of the compoets of X Lemma. For a fixed iteger > 0-, there are at most a fiite umber of D i such that IU(K(L,i),e) I D i ad IU(K(R,i),e ) I D i are ot cotaied i the same compoet of Iu(x,e ) ID.. 1 The lemma is trivially true if X cosists of a fiite umber of compoets. Thus the lemma is oly iterestig i the case where X cosists of a ifiite umber of compoets. Suppose the lemma is ot true for D ifiite. I other words, suppose there exists a positive iteger N such that

4 TOPOLOGY PROCEEDINGS Volume there exist a ifiite umber of D such that IU(K(L,i),eN) I D i i ad IU(K(R,i),eN) I D i are ot cotaied i the same compoet of lu(x,e N ) I D i. Let {Di,Di'... } be such a ifiite subset of D. Let K' (L,i) ad K' (R,i) be the respective left ad right compoets of X i 0i' i = 1,2,... Furthermore let x' (L,i) be a arbitrary poit of K' (L, i). The the set {x' (L, i) }~=l ' cotais a limit poit x' (L) sice X is compact. Let G(L) be the compoet of X cotaiig x' (L). It follows that for every positive iteger, B(x' (L),e) itersects ifiitely may x'(l,i) E {X'(L,j)}j=l. Choose x"(l,l) from {X'(L,i)}~=l such that x" (L,l) E B(x' (L),e ), ad x" (L,) from {x' (L,i) }:=l l \{X"(L,i.)}~-ll such that x"(l,) E B(x'(L),e) ad 1= d(x'(l),x"(l,-l» >d(x'(l),x"(l,», = 2,3,... We ow defie Di such that x"(l,i) E Dr ad Di E {Di,Di'... } for all positive itegers i. Also K"(R,i) is defied to be the right compoet of D~ X, ad x"(r,i) is defied to be a arbi 1 trary poit of K"(R,i) for all positive itegers i. The set of poits {X"(R,i)}:=l is a ifiite set of poits belogig to the compact space X. Hece the set of poits has a limit poit, say x" (R), where x" (R) E X. Thus for every positive iteger, B(x"(R),e ) itersects ifiitely may x"(r,i) E {X"(R,j)}j=l. Choose x 111 (R,1) E {X"(R,J')}~ 1. Also choose x"'(r,) E B(x"(R),e) J= satisfyig x '" (R, ) E {x" (R, j) };=1\ {x '" (R, j) }j:i, ad such that x'" (R,-l) = x" (R,j-l) ad x II (R,) = x" (R,j) implies j-l < j for all itegers greater tha oe. Defie D~II E {D'!}c:' such 1 J J=l that x'" (R,i) E Di". It follows that for all itegers N' greater tha N; there exists a positive iteger m satisfyig the iequalities d(x' (L),x'" (L,m» < lin' ad d(x" (R),x'" (R,m» < lin'. It is ot hard to show that x' (L) ad x"(r) have the same first coordiate. Thus if G(R) is the compoet of X such that x"(r) belogs to G(R), the G(R) = G(L). This implies B (x' (L),eN) IU (K ". (L, m),en) I D~" t- ~,

5 94 Laidacker ad B (x" (R), en) Iu (K "' (R,m), en) I D~" t- ~. Cosequetly \U(K"' (L,m),eN) I D~" ad IU(K'" (R,m),eN) I D"' m are cotaied i the same compoet of the ope set lu(x,e N ) I D~'. This is a cotradictio sice D~' belogs to {Di}~=l. For the ext two lemmas, we use the followig otatio. Let I' be a closed iterval i the iterval [-1,2]. Defie X' to be X (I' x Q) ad D(I') to be the set of compoets of (I' x Q)\(C o x Q) Lemma. For a fixed iteger > 0" there are at most a fiite umber of D. such that IU(K(L,i),e) I D i ad 1 IU(K(R,i)e )1 D. are ot cotaied i the same compoet of 1 IU(X',e ) I D. for D. E D(I'). 1 1 The above lemma is the same as 2.1 oly with a otatioal chage Lemma. If N is a fixed positive iteger ad for all D i i D(I'), IU(K(L,i),eN)1 D i ad IU(K(R,i),eN)1 D i are cotaied i the same compoet of IU{X',eN) I D i " the IU(X',eN) I (I' x Q) is a cotiuum cotaiig X'. The proof is left to the reader Lemma. Let N be a fixed positive iteger" D be a i elemet of D" ad K i be a compoet of X i D i. The i IU(Ki,eN)1 D i there is a pseudocoe S over K i such that (bdry Pi) S = Ki The proof is essetially the same as that preseted for Lemma 3 of [1] Lemma. Let N be a fixed positive iteger ad D be i such that I U(K(L,i),eN) 1 D i ad I U(K(R,i),eN) I D i are

6 TOPOLOGY PROCEEDINGS Volume co taied i t he same compoet of Iu (X', en) I D i. is a homeomorphism hi: (-00,00) -+ [IU (X', en) I D i ] such that hi((-oo,oo)) = K(L,i) U h((-oo,oo)) U K(R,i). The there Lemma 2.4 is used to get two disjoit pseudocoes, oe over K(L,i) ad the other over K(R,i). This is doe so that the uio of the pseudocoes itersected with the boudary of D i is K(L,i) U K(R,i). It is a simple matter to joi the two vertices of the pseudocoes with a arc i the compoet of lu(x,e N ) I D i cotaiig IU(K(L,i),eN) I D i ad IU(K(R,i),eN) I D i such that the arc mius its ed poits does ot itersect the two pseudocoes. I the future, we will refer to h((-oo,oo)) as a coector Theorem. Give ay compactum X as described i sectio l~ there is a cotiuum I(X) such that two compoets of I(X)\X are homeomorphic to half ope itervals ad the remaiig compoets are homeomorphic to ope itervals. Furthermore I (X) = I (X)\X, ad for c E [-1,2]\Co~ {c} x Q itersects at most oe of the two half ope itervals or oe of the ope itervals. It does ot~ however~ itersect a half ope iterval ad a ope iterval. I D l ad D 2, we costruct two pseudocoes over K(R,l) ad K(L,2) as described i 2.4. If X cotais oly a fiite umber of compoets, the i each of the remaiig D i E D, we costruct coectors as described i 2.5. The uio of X, the two pseudocoes, ad the coectors, is a cotiuum satisfyig the theorem. Suppose X cosists of a ifiite umber of compoets. We first recall that e 4 ad for x,y E X, d (x,y) I l fact the above iequality holds for x,y E I x Q where I = [0,1].

7 96 Laidacker We ote oe other thig, amely that /u(x,e l ) / (I x Q) = I x Q. Thus the cotiuum Ai, cosistig of the two pseudocoes uioed with I x Q, cotais x. We will defie A iductively. Suppose A is a cotiuum - l i M x Q such that A - l cotais two pseudocoes, as described i 2.4, i D ad D 2. Also A - cotais a fiite umber of l l coectors as described i 2.5. Let D~,,D~ be the elemets -l of D correspodig to the coectors of A -1. Let Il,...,I k -l be the largest closed itervals i I such that I j x Q does ot itersect a coector. We assume that for N = i - l ad D i i (II x Q) U... U (I k x Q) where D i E D, the hypothesis of 2.5 -l is satisfied. I additio, suppose A - l cotais the compoets of Iu(x,e. ) I (I. x Q) which cotai X (I. x Q). Call I - l J J these compoets H, j j = l,...,k _. The cotiuum A - is l l the uio of two pseudocoes, a fiite umber of coectors, ad cotiua Hl,...,H as described above. Furthermore, two of k -l the compoets of A_l\(H l U... U H ) are homeomorphic to k -l half ope itervals while the remaiig compoets are homeomorphic to ope itervals. We defie A - l = AI' ad e. = e I - l l for = 2. We will ow defie A assumig A - 1 is kow. We choose a iteger i greater tha i - such that for at l least oe D i E D, D i (I. x Q) ~~, IU(K(L,i),e ) I D ad J i i IU(K(R,i),e. ) I D i are ot cotaied i the same compoet of l Iu(x,e. ) I (I. x Q) where j E {1,2,...,k _ }." By 2.2, there l J l are at most a fiite umber of D E D, say D~ +l,...,d~, i -l with the above property. If a E {d_l+l,d_l+2,...,d}' the there is a j E {l,...,k _ } such that D~ (I x Q) ~~. For l j each such D' there is by 2.5 a coector i the compoet of Iu(x,e ) i I D' cotaiig IU(K(L,i),e. ) I D' ad -l a I - l a /U(K(R,i),e. ) I D'. These ew coectors are i I - l a HI U... U H. Let Ii,...,I k k be the largest closed i-i tervals i I such that I! x Q does ot itersect a coector J

8 TOPOLOGY PROCEEDINGS Volume of A-I' a ew coector, or the half ope itervals of the pseudocoes. Thus for N = i ad 0i i (Ii x QJ U U (I~ x QJ where D i E D, the hypothesis of 2.5 is satisfied. Let A cotai the compoets of IU(X,e. ) I (I~ x Q) which cotai X (I~ x Q), l J J j = 1,2,...,k. The the cotiuum A is the uio of two pseudocoes (the oes from A-I)' a fiite umber of coectors (the oes from A - plus the ew coectors defied above), ad l cotiua Hi,...,H where H~ is the compoet of Iutx,e. ) k I J l (Ij x Q) metioed above, j = l,...,k. The cotiuum A has the property that two of the compoets of A\(Hi U U K k ) are homeomorphic to the half ope itervals while the remaiig compoets are homeomorphic to ope itervals. The half ope itervals of the pseudocoes of A ad the coectors of A are such that for c E [-1,2]\C ' if {c} x Q o itersects a coector, the {c} x Q itersects at most oe coector ad does ot itersect a half ope iterval of a pseudocoe of A. If {c} x Q itersects a half ope iterval of a pseudocoe of A' the {c} x Q itersects oly oe of the half ope itervals of the pseudocoes ad does ot itersect ay of the coectors of A. Oe of the properties of A' 1,2,..., is Al :::> A 2 ::::> A 3 ::::> We defie I(X) = ~=IAi. It is obvious that I(X) has the properties described i Theorem 2.6. We will call the half ope iterval of the left pseudocoe, the left coectop ad the half ope iterval of the right pseudocoe, the pight coectop. 3. Properties of I (X) relative to the compoets of X We show i this sectio that I(X) preserves may properties possessed by the compoets of X provided all of the compoets of X have these properties.

9 98 Laidacker We first state the theorem o which all of the remaiig results of this sectio deped Theorem. There is a mootoe mappig f of I(X) oto M such that f: (I(X)\X) ~ (M\C ) is a homeomorphism ad poit o iverses of Co are compoets,of X. The proof of this theorem is straightforward. I fact, the proofs of may of the followig theorems are straightforward ad thus are left to the reader Theorem. If X is ay compactum~ the I(X) is irreducible betwee two of its poits. We list a few defiitios that will be used i the theorems which follow. We say a cotiuum X is acyclic if every mappig of X ito the circle i homotopic to a costat mappig. Give that each collectio of mutually disjoit odegeerate subcotiua of a cotiuum X is coutable, we say that X is Susliia. A cotiuum is ratioal if it has a basis of ope sets with coutable boudaries. If a is a ope cover of the compactum X, a map f of X oto a compactum Y is called a a-map provided that for each y i Y, f-l(y) is cotaied i some member of a. A tree is a I-dimesioal acyclic coected graph. A cotiuum X is tree-like if for each ope cover a of X, there is a a-map of X oto some tree. Arc-like is defied i a similar maer Theorem. If X is ay compactum~ the I(X) is decomposable Theorem. If X is a compactum ad each odegeerate compoet is hereditarily decomposable~ the I(X) is hereditarily decomposable Theorem. If X is a compactum~ the I(X) is uicoheret.

10 TOPOLOGY PROCEEDINGS Volume By 3.3, I(X} is decomp6sable. If A ad Bare two proper subcotiua of I(X} such that I(X} = A U B, the either A or B must cotai the ed poit of the left coector. Without loss of geerality, we assume A cotais the ed poit. The f(a} cotais the poit -1. It follows that f(b} cotais 2, but ot -1. Sice A is coected ad otrivial, f(a} is a iterval [-l,a] i M. For the same reasos f(b} is a iterval [b,2] i M. Thus f(a B} = [b,a] or A B = f-l([b,a]), either of which is a cotiuum. For A or B a poit, the proof is trivial Theorem. If X is a compactum ad each compoet of X is hereditarily uicoheret~ the I(X) is hereditarily uicoheret. The proof is similar to the proof of Theorem. If X is a compactum ad each odegeerate compoet is a ~-dedroid~ the I(X) is a ~-dedroid. Sice a ~-dedroid is by defiitio a hereditarily uicoheret, hereditarily decomposable cotiuum, 3.7 follows directly from 3.4 ad Theorem. If X is a compactum ad each compoet is atriodic~ the I(X} is atriodic Theorem. If X is a compactum ad each compoet is either a poit or a arc-like cotiuum~ the I(X) is a arclike cotiuum. Notice that each compoet of X is hereditarily uicoheret ad atriodic. Thus by 3.6 ad 3.8, I(X} is hereditarily uicoheret ad atriodic. If a odegeerate subcotiuum Y of I(X} is idecomposable, the Y is a subcotiuum of a compoet of X. I this case Y is arc-like. Thus I(X} is hereditarily uicoheret, atriodic, ad every otrivial

11 100 Laidacker idecomposable subcotiuum of I(X) is arc-like. By Theorem 2 of [9], I(X) is arc-like. Remarks. The reader should compare 3.9 with Theorem 2.1 of [16], ad Theorem 11 of [5]. Theorem 3.9 aswers the questio posed by A. Lelek i a coversatio with the author, "Ca every compactum whose compoets are arc-like be imbedded i a arc-like cotiuum?" Corollary. If X is a compactum ad each odegeerate compoet of X is arc-like~ the X is plaar. is plaar. Big [3] has proved that every arc-like cotiuum This ad 3.9 yield the corollary Theorem. If X is a compactum ad each compoet is either a poit or a tree-like cotiuum~ the I(X) is a tree-like cotiuum. Agai we ote that each compoet of X is hereditarily uicoheret. Thus I(X) is hereditarily uicoheret by 3.6. If Y is a otrivial idecomposable subcotiuum of I(X), the Y is a subcotiuum of a compoet of X. Sice every subcotiuum of a tree-like cotiuum is tree-like, Y is tree-like. Hece I(X) is hereditarily uicoheret ad every otrivial idecomposable subcotiuum of I(X) is tree-like. By Theorem 1 of [6], I(X) is tree-like Theorem. If X is a compactum ad each compoet is of dimesio q.t most k (k > 0) ~ the I (X) is of dimesio at most k. The cotiuum I(X) is the uio of X ad I(X)\X. The set I(X)\X cosists of a coutable umber of compoets each of which is homeomorphic to either a ope iterval or a half ope iterval. Thus I(X)\X is the coutable uio of arcs. We ote that X is of dimesio at most k (k > 0). From

12 TOPOLOGY PROCEEDINGS Volume Theorem 1112 of [10], we get our theorem Theorem. If X is a compactum ad each compoet is acyclic, the I(X) is acyclic. Let g be a mappig of I(X) ito the circle. The mappig is either essetial or it is ot. If the mappig is ot essetial, the it is iessetial ad i this case is homotopic to a costat mappig. If g is a essetial mappig of I(X) ito the circle sl, the there exists i I(X) a cotiuum K with the property that glk is ot homotopic to a costat mappig, but every proper subcotiuum K I of K is such that glki is homotopic to a costat mappig. Furthermore, K is discoheret ([4], p. 216). The cotiuum K is discoheret if the complemet of each subcotiuum of K is coected (refer to [15], p. 163). If f(k) is ot a poit, the K is uicoheret. This we have proved earlier. Thus K must be a subcotiuum of oe of the compoets of X. Sice glk is ot homotopic to a costat mappig, g restricted to the compoet N of X cotaiig K is ot homotopic to a costat mappig. It follows that N is ot acyclic. This cotradicts the fact that N is acyclic. Sice g is ot a essetial mappig, g is a iessetial mappig. Also sice g was arbitrary, we have that I(X) is acyclic Theorem. If X is a compactum where each compoet is ratioal, ad X cotais at most a coutable umber of odegeerate compoets, the I(X) is ratioal Theorem. If X is a compactum where each compoet is Susliia, ad X cotais at most a coutable umber of odegeerate compoets, the I(X) is Susliia. Remarks. There might be times whe oe would wat to imbed

13 102 Laidacker a compactum as described i the hypothesis of 3.15 i a locally coected Susliia curve. Fitzpatrick ad Lelek describe such a imbeddig i [8]. It follows from their work that subcompacta of Susliia cotiua are characterized by those properties described i the hypothesis of Characteristics of subcompacta of specific curves Recall that a curve is a I-dimesioal cotiuum. I this sectio we iclude some of the more immediate results which follow from our work i sectio Theorem. A compactum X ca be imbedded i a ratioal curve if ad oly if X cotais at most a coutable umber of otrivial compoets ad each compoet is ratioal. This theorem follows directly from 3.14 ad Theorem 5, page 285, i [15] Theorem. A compactum X ca be imbedded i a acyclic curve if ad oly if the compoets of X are either degeerate or acyclic curves. If Y is a acyclic curve, the every subcotiuum of y is a acyclic curve. By 3.12 ad 3.13, if every compoet of X is acyclic ad at most I-dimesioal, the X ca be imbedded i a acyclic I-dimesioal cotiuum Theorem. A compactum X ca be imbedded i a atriodic tree-like curve if ad oly if each odegeerate compoet of X is atriodic ad tree-like. The proof follows immediately from 3.8 ad Corollary. Either there exist oplaar atriodic tree-like curves 3 or give ay atriodic tree-like curve X 3 the plae cotais ucoutably may mutually disjoit homeomorphic

14 TOPOLOGY PROCEEDINGS Volume copies of x. Let X be a atriodic tree-like curve. Either X is oplaar or it is ot. If it is oplaar, the the theorem is true. If X is plaar, the C x X, where C is the usual Cator set, is a compactum ad each compoet is atriodic ad tree-like. Hece C x X ca be imbedded i a atriodic treelike curve Y by 4.3. Either Y is plaar or it is ot. If Y is plaar, the the plae cotais ucoutably may mutually disjoit homeomorphic copies of X. If Y is ot plaar, the the first part of the theorem is true. Remarks. This result relates two questios. Big i [3] asks if give ay atriodic tree-like plaar curve X, does the plae cotai ucoutably may mutually disjoit homeomorphic copies of X? Igram [11] proved the existece of a atriodic tree-like curve i the plae which is ot arc-like. Furthermore i [12), Igram proved that ucoutably may atriodic tree-like cotiua, oe of which is arc-like, ca be imbedded i the plae so that they are mutually exclusive. Igram, at this coferece, asked whether there exist oplaar atriodic treelike curves Theorem. A compactum X ca be imbedded i a hereditarily decomposable cotiuum if ad oly if each odegeerate compoet of X is hereditarily decomposable. From 3.4 ad the defiitio of hereditarily decomposable, the theorem easily follows Theorem. A compactum X ca be imbedded i a hereditarily uicoheret curve if ad oly if each odegeerate compoet of X is a hereditarily uicoheret curve. Theorem 4.6 follows directly from 3.6 ad the defiitio of hereditarily uicoheret.

15 104 Laidacker 4.7. Theorem. A compactum X ca be imbedded i a A-dedroid if ad oly if each odegeerate compoet of X is a A-dedroid. This theorem follows directly from 3.7 ad the defiitio of A-dedroid. Remarks. It is evidet that we did ot exhaust the possible results i this sectio. However, it is clear how oe would proceed i order to get similar results. Bibliography 1. D. B. Bellamy, A o-metric idecomposable cotiuum, Duke Mathematical Joural 38 (1971), , A ucoutable collectio of chaiable cotiua, Trasactios of the America Mathematical Society 160 (1971), R. H. Big, Sake-like cotiua, Duke Mathematical Joural 18 (1951), J. J. Charatoik, Cofluet mappigs ad uicoherece of cotiua, Fudameta Mathematicae 56 (1964), H. Cook, O the most geeral plae closed poit set through which it is possible to pass a pseudo-arc, Fudameta Mathematicae 55 (1964), , Tree-likeess of dedroids ad A-dedroids, Fudameta Mathematicae 64 (1970), , ad A. Le1ek, O the topology of curves IV, Fudameta Mathematicae 76 (1972), B. Fitzpatrick Jr.,ad A. Le1ek, Some local properties of Susliia compacta, Colloquium Mathematicum 31 (1974), J. B. Fugate, A characterizatio of chaiable cotiua, Caadia Joural of Mathematics 21 (1969), W. Hurewicz, ad H. Wa1ma, Dimesio Theory, Priceto Uiversity Press, Priceto, W. T. Igram, A atriodic tree-like cotiuum with positive spa, Fudameta Mathematicae 77 (1972), , A ucoutable collectio of mutually exclusive atriodic tree-like cotiua with positive spa, Fudameta Mathematicae 85 (1974), J. R. Isbell, Embeddigs of iverse limits, Aals of

16 TOPOLOGY PROCEEDINGS Volume Mathematics 70 (1969), K. Kuratowski, Topology I, Academic Press, New York, , Topology II, Academic Press, New York, A. Le1ek, O the topology of curves I, Fudameta Mathematicae 67 (1970), Lamar Uiversity Beaumot, Texas 77710