 Advances in Pure Mathematics, 2012, 2, 133-138 http://dx.doi.org/10.4236/apm.2012.22020 Published Online March 2012 (http://www.SciRP.org/journal/apm) Making Holes in the Hyperspace of Subcontinua of Some Continua José G. Anaya, Enrique Castañeda-Alvarado, Fernando Orozco-Zitli Facultad de Ciencias, Universidad Autónoma del Estado de México, Toluca, México Email: {jgao, eca}@uaemex.mx, forozcozitli@gmail.com Received November 25, 2011; revised December 23, 2011; accepted December 30, 2011 ABSTRACT ACX, Let X be a metric continuum. Let A is said to make a hole in CX, if is not unico-herent. In this paper, we characterize elements ACXACX such that A makes a hole in , where CXX is either a smooth fan or an Elsa continuum. Keywords: Continuum; Elsa Continuum; Fan; Hyperspace; Property b); Unicoherence; Whitney Map 1. Introduction A connected topological space Z is unicoherent if when- ever =ZAB, where A and are connected and closed subsets of BZ, the set AB is connected. Let Z be a unicoherent topological space and let be an element of zZ. We say that makes a hole in zZ if Zz is not unicoherent. A compactum is a nonde- generate compact metric space. A continuum is a con- nected compactum with metric . Given a continuum dX, the hyperspace of all nonempty subcontinua of X is denoted by and it is considered with the Haus- dorff metric. It is known that the hyperspace CXCX is unicoherent (see [1, Theorem 19.8, p. 159]). In the papers  and  the author present some par- tial solution to the following problem. Problem. Let X be a hyperspace of X such that X is unicoherent. For which elements, AX, does A make a hole in X? In the current paper we present the solution to that pro- blem when X is either a smooth fan or an Elsa con- tinuum and  =XCX. 2. Preliminary We use and to denote the set of positive integers and the set of real numbers, respectively. Let Z be a topological space and let A be a subset of Z. We denote intA the interior of A in Z. An arc is any homeomorphic space to the closed unit interval . Let in a topological space [0,1],pqZ, ,pq will denote an arc, where and are the end points of p q,pq . A free arc in a continuum X is an arc ,pq such that pq p,q, is open in X. A point in a connected topological space Z is a cut point of (non-cut point of) Z provided that Zz1: is disconnected (is connected). A map is a continuous function. A map fZS, where Z is a connected topological space and is the unit circle in the Euclidean plane , has a lifting if there exists a map such that 1S2:hZ=expfhexp 1S, where is the map from onto defined by exp=cos 2π,sin2πttt. A connected topological space 1:Z has property b) if each map fZS has a lifting. By an end point of X, we mean an end point in the classical sense, which means a point of pX that is a non-cut point of any arc in X that contains . A sub- space of a topological space pYZ is a deformation retract of Z if there exists a map H:ZI Z such that, for each zxZ, ,0 =xx, 1=HHZY and, for each ,1 =yYH, yy zZ. We say that a topolo- gical space Z is contractible if there exists , such that z is a deformation retract of Z. It is known that each contractible normal topological space has property b), and so it is unicoherent (see [4, Theorems 2 and 3, pp. 69 and 70]). 3. Smooth Fans A point of a continuum pX is a ramification point provided that is a point which is a common end point of three or more arcs in pX that are otherwise disjoint. A fan is an arcwise connected, hereditarily unicoherent continuum with exactly one ramification point (here- ditarily unicoherent means each subcontinuum is uni- coherent). The ramification point of a fan will be called the vertex of the fan. If X is a fan and xyX, then ,Copyright © 2012 SciRes. APM J. G. ANAYA ET AL. 134 Copyright © 2012 SciRes. APM ,xy denotes the unique arc joining x and . A fan yX with vertex v is said to be smooth provided that if is a sequence in nnx=1X such that it converges to a point xX=1,nnvx , then the sequence converges to ,vx in CX. To establish some notation, let X be a smooth fan with vertex and let i be its end- points set, where is an infinity indexing set. It fol- lows from definition of smoothness that the set: vEX=:iex X=,Xv:xNC is a natural homeomorphic copy of X in CX. By the smoothness of X, we have that the set: =,C vieiXCXTC is a closed subspace of . Furthermore, each hy- perspace ,Cvie is a 2-cell and ,=ejv,viCvCve for each which ,ijCX,CvXare different. The set of all elements of such that it contains will be denoted by . Let ACX. We say that A is a simple arc if A is an arc such that and, there exists a sequence of satisfying the following properties: X=AECX nAim=1n1) =l nAAnnAnAAA and 2) for each , vint a) , b) and c) . nSince X is embeded in the Cantor fan (see ), we can regard X as embedded in the Euclidean plane such that v and each i is a convex arc, where . Note that for , i for each i. Throughout this section will de- note the map from 2]i=re veEXh=0eE,0X[,ve=0r0,1X onto TC defined by X,=tx,hxt x. We assume in this section that if ,,ieabv , then the distance between and v abis less than the distance between and . vLemma 3.1. Let X be a smooth fan with vertex . If v,abieE is an arc contained in , where , then: 0iX,ve01) If ,b=1nnx=int a, there exists a sequence  of X such that ,=mvb vli ,nx and, for each n, 0,nixve. 2) If 0i and ,aeb,binta, then 0,ibe is a free arc in X. Proof. The proof of (1) is easy. In order to prove (2), we suppose that is not a free arc in 0,ibeX. Then there exists 00,,iie be0yb such that 00,iyintbe. Hence, 00,iyintve=1nny0,i. Then, there exists a sequence  of Xve0=lim nyy such that . Since X a smooth fan, 0,=lim,nvy vy. Notice that 0,,,ab vbvy . Let ,zintab=1nnz0. There exists a sequence  of X such that 0 and, for each n, =lim nzz ,nzvyn. Clearly ,vintab0zv. Hence, . Let >0 be such that vBz0 and 0,Bz ab0n. Let 000,,nizBz abve be large enough such that 00 0,,ni nzve vy. Thus,   . Since X is a fan and 00,niyXve0=nzvQ1,1, 1,QqQ, , this is a contradiction. □ Since the Hilbert cube, , is homogeneous (see [1, Theorem 11.9.1, p. 93]) and is con- tractible, we have the following result. Lemma 3.2. Let . Then has property b). QqTheorem 3.3. Let X be a smooth fan with vertex . If vA is a subcontinuum of X such that vA and, for each eEXi, ,iAve, then A does not make a hole in CX. Proof. We are going to prove that ,CvX ACX A is a deformation retract of . Notice that, for each BTCX, there exists ,0,1BBxt X such that ,=BBhx tB  ,if ,,=, 1,,if .BBBBBCvXHBt ttxx BTCX. We define  Clearly H is a map. Then, ,X ACv is a deformation retract of CX A. Since Q is ho- meomorphic to ,CvX (see [6, Theorem 3.1, p. 282]), {},CvX A has property b) (see Lemma 3.2). Therefore CX A has property b) (see [2, Propo- sition 9, p. 2001]). □ Lemma 3.4. Let X be a smooth fan with vertex vand let 0,,iabCve be a simple arc contained in X, for some 0ieEX. Then 00,=,iiintb eb eb. Proof. Since ,abA is a simple arc, there exists a sequence =1nn of CX n that satisfies the required properties of the definition. Notice that, for each , nATCX0,ni and Ave. Given nna0,nibve, let ,  such that =,nnnAab. We need to prove the following claim. J. G. ANAYA ET AL. 135Claim. is a free arc in 0i,beX. Let . First, we suppose that there exists n0n such that 0nb00,nnb,vb int a. Since 0nb00,nibe0,ibe and 00,inea, is a free arc (see (2) of Lem- ma 3.1). Hence, is a free arc in X. Now, we assume that, for each , n,nibb00,,iie be,>0y0n0e b yb. Let . Notice that =lim nbb and db. Then there exists 0,0idbe0=limimey mm0,miybe,=,ive va00, ,ab be, and 0, we may assume that, for each e, . Since 0,miyvX is a smooth fan, lim,imvy 0,,zbe0ve,=00 0,ii ibe ve. Let mz0=limmzz. There exists a sequence such that, =1mand, for each , m,zvy00,,iibe bemm is an open set in . Since X and 00,,iie be 0m0ibe ,,=mvyv,iib eb0zbzb, there exists such that . Then 00,,mie 0,ive00mizveint, this is a contradiction. Therefore . □ ,=be00Theorem 3.5. Let X be a smooth fan with vertex v. If ACX is a simple arc, then A makes a hole in . CX0=, ,iAabCveProof. We may assume that , where 0ieEX and 00,1t0=atb such that . Let: 0=(,,iCvXTCXintbeA  and 0=, 0,1ihbeA0,iXintbe. By Lemma 3.4, is a smooth fan. Then 0,iTCX int beA is a connected and closed subset of CX ACX A. So, is a con- nected and closed subset of . Notice that is homeomorphic to 00,0,1,ibe bt. Since 00,0,1,ibe bt is a connected subset of 00,1 ,XbtCX A, we have that is a connected subset of . Clearly is a closed subset of CX A. ,=CX abNotice that  and: 00=0,1, ,0ihbbt hbe. Let: 1000=0,, ,0ihb tbthbe  and 200=,1,hb tbt12=. Clearly, is a separation of . Then CX Av is not unicoherent. □ Theorem 3.6. Let X be a smooth fan with vertex , let 0ieEX00,iiavee0,iae and let . Then CX does not make a hole in . Proof. Let 00:,[0,1],iiGCX aeCX ae be defined by: ,= :GAttaa AGG. It is easy to prove that is well defined. In order to show that is continuous, we define :0,1GCX CX by ,= :GAttaa AG=1,nn nAt . We prove that is continuous. Let be a se- quence in 0,1CX and 00,0,1At CX such 00,=lim ,Athat nntAt. We suppose that there exists BCX=lim ,BGAt such that nn. We will show =,BGAtbB00 . Let . Consider two sequences =1nnb and =1nna of X such that Copyright © 2012 SciRes. APM J. G. ANAYA ET AL. 136 =limb,bGAtaXnn00,AtBG0 0,GA tA=1nna:0,1H by n and, for each , nnn, nn and nn. Taking subsequences if nece- ssary, we may assume that there exists 0 such that . Then . Moreover, b=nim na=ta bnbta0aAaA0=la=limta 000 and, so bG . This proves that . Now, let 00 . Then 00. Then there exists a sequence 00,At taa in X such that and, for each , 0nnaA=lim naa n. So . Since, for each , , . Thus . 00tannta =liGmta=,Atnn,nAGntta Bn00 00Hence, is a map. So BGv is a deformation retract of . 0,iae0,iae0eCX Then is contractible. Therefore has property b) (see [2, Proposition 9, CX,iCX ap. 2001]). □ Theorem 3.7. Let X be a smooth fan with vertex , v0ieE0,,iC velet and let  such that Xab0i,eab and is not a free arc of 0,ibeX. Then ,abCX does not make a hole in . Proof. In light of Proposition 9 of [2, p. 2001], it suffices to prove that there exist two connected, closed subsets and  of ,abCX which have pro- perty b) and the intersection of them is connected. We may assume that there exists 00,1t0=tb a0=0t such that . We consider two cases. Case 1. . Then ,=,vbab . Let =,XabTC and ,X ab=,Cv. Clearly has property b). By Theorem 3.1 of [6, p. 282], is a Hilbert cube. By Lemma 3.2, ,CvX has property b). Notice that ,X ab,=abN C. Clearly ,XabNC is homeomorphic to Xb. Since Xb is connected, ,ab is con- nected. By Proposition 8 of ,  ) ,ab[,]=(CX ab0>0t has property b). Case 2. . Consider the following sets: 001,bt=,hX t and 000, ,tbt=,CvX hX . Clearly and  are connected, closed subsets of ,abCX  and 00,t bt=hX. Notice that  is homeomorphic to Xb. So, since Xb is connected,  is connected.  have property b). If we define Now, we are going to prove that and ,, =,1Hhxt shxtts, we have 1hX1hX is a deformation retract of . Since is con- tractible, 1hX has property b) (see [2, Proposi- tion 9, p. 2001]). Hence, has property b) (see [2, Proposition 9, p. 2001]). In order to prove that  has property b), note that ,CvX is a deformation retract of . By Theorem 3.1 of [6, p. 282], ,CvX is homeomorphic to a Hilbert cube. Thus, ,CvX has property b). Hence,  has property b) (see Proposition 9 of [2, p. 2001]). Therefore [,]=CX ab has property b). □ Classification Theorem 3.8. Let X be a smooth fan with vertex and vCX. Then A makes a hole in ACX if and only if A a simple arc. Proof. Let CX be such that AA makes a hole in CX. By Theorem 3 of [2, p. 2001] and by Theorem 3.3, A is an arc ,pq. By Theorems 3.6 and 00,, ,iipq veve for some 3.7, 0iX0,iqeeE , and X. In order to prove that  is a free arc in ,pq=1nnanb is a simple arc, let , be sequen- =1nces in ,,pq pq0,iqe q=limpa =limb and , respectively, such that and q. Then n n,= ,nnpqlima bn and, for each , ,vabnn, ,inta bnn,,nnpqa b. Therefore  and A is a simple arc. The sufficiency follows from Theorem 3.5. 4. Elsa Continua A compactification of 0,0, with an arc as the re- mainder is called an Elsa continuum. The Elsa continua was defined by S. B. Nadler Jr., in . A particular example of an Elsa continuum is the familiar sin(1/x)- continuum. There are uncountably many topologically different Elsa continua, the different topological types being a consequence of different ways “patterns into” the remainder of the compactification [8, p. 184]. Let X be a continuum. A Whitney map for CX is a continuous function :[0,1]CX that satisfies the following two conditions: BCX such that A, AB and 1) for any 