On the planarity of Peano generalized continua: An extension of a theorem of S. Claytor

C O L L O Q U I U M M A T H E M A T I C U M VOL. 75 1998 NO. 2 ON THE PLANARITY OF PEANO GENERALIZED CONTINUA: AN EXTENSION OF A THEOREM OF S. CLAYTOR BY R. A Y A L A, M. J. C H ´ A V E Z AND A. Q U I N T E R O (SEVILLA) We extend a theorem of S. Claytor in order to characterize the Peano generalized continua which are embeddable into the 2-sphere. We also give a characterization of the Peano generalized continua which admit closed embeddings in the Euclidean plane. 1. Introduction. The celebrated Kuratowski Theorem [7] states that a finite graph Gis embeddable in the 2-sphere S2if and only if Gcontains no subgraph homeomorphic to the complete bipartite graph K3,3or to the complete graph with five vertices K5(see Fig. 1). ' & $ % @@@s s ss ss' & $ % @@@    AAA A sss s s K3,3≡≡K5 Fig. 1 Later S. Claytor [2] characterized Peano continua which are embeddable in S2by adding to K3,3and K5two further forbidden curves L1and L2 which are non-polyhedral 1-dimensional Peano continua constructed from K3,3and K5respectively (see Fig. 2). @@@@s s ss ss @@@s s ss ss@@ssss s s s p1 qqqqΣ1 ≡L1 ≡L2 @@@@    AAAAsss s s @@@    AAA A sss s s @@  AAsss s s s p2 qqqqΣ2 Fig. 2 1991 Mathematics Subject Classification: 54C10, 54F15. [175] 176 R. AYALA ET AL. Actually, L1and L2are due to Kuratowski who already suggested a role for them in his paper [7]. The Kuratowski Theorem was extended to any locally finite graph Gby Dirac and Schuster [3] by proving that Gis planar if and only if all its finite subgraphs are planar. In addition, R. Halin [6] characterized the locally finite graphs which admit planar embeddings without vertex accumulation points (VAP-free embedding) by six forbidden graphs. Namely K5, K3,3, and the four infinite graphs in Fig. 3 below. Later C. Thomassen ([10]; Cor. 4.1) showed that for connected locally finite graphs VAP-free embeddings are the same as closed embeddings. Also two-dimensional finite complexes embeddable in S2have been characterized by Mardeˇsi´c and Segal in [8]. The authors use Claytor’s Theorem to show that K5, K3,3and the space F⊆R3called the “spiked disk” given by F={(x, y, 0) : x2+y2≤1} ∪ {(0,0, z):0≤z≤1}are the forbidden subspaces for the planarity of finite 2-complexes. In this paper we show that Claytor’s Theorem still holds for Peano generalized continua. See §2 for definitions. Namely we prove Theorem 1.1. Let Xbe a Peano generalized continuum. Then the following statements are equivalent: (1) Xis embeddable in S2(or equivalently in R2if X6=S2). (2) Any subcontinuum K⊆Xembeds in S2. (3) Xcontains no set homeomorphic to K5, K3,3, L1, L2. (4) The Freudenthal compactification b Xof Xis embeddable in S2. We also characterize the Peano generalized continua which admit closed embeddings in R2. More explicitly, a Peano generalized continuum Pis said to be properly planar if there exists a proper (or equivalently closed) embedding of Pinto the Euclidean plane R2. See §2 for definitions. K∞ 5≡ qqq qqq qqq qqq @@@@ sss sss sss sss L∞ 3,3≡ qqq qqq@@@@ssss sssss s K∞ 3,3≡ qqq qqq qqq @@@@ sss s ssss sss L∞ 5≡ qqq qqq @@@@    AAAAsssssss s s Fig. 3 Theorem 1.2. Let X6=S2be a Peano generalized continuum. Then the following statements are equivalent: PLANARITY OF CONTINUA 177 (a) Xis properly planar. (b) The one-point compactification X+is embeddable in S2. (c) Xcontains no closed set homeomorphic to K5, K3,3, L1, L2or any of the four Halin graphs of Fig. 3. These results are extensions to generalized continua of the quoted Dirac– Schuster Theorem and Halin–Thomassen result respectively. Moreover, we derive from Theorems 1.1 and 1.2 a characterization of (properly) planar locally compact 2-polyhedra. Actually the characterization of planar compact polyhedra in [8] can now be extended to locally compact polyhedra with small changes. Namely, if one uses 1.1 instead of Claytor’s Theorem in the proof of (c)⇒(a) in ([8]; §3) one gets Corollary 1.3. A locally compact two-dimensional polyhedron Pis embeddable in S2if and only if Pcontains no set homeomorphic to K3,3, K5 or the spiked disk F. Similarly by using 1.2 one gets Corollary 1.4. A locally compact two-dimensional polyhedron P6=S2 is properly planar if and only if Pcontains no closed set homeomorphic to K3,3, K5, K∞ 5, L∞ 5, K∞ 3,3, L∞ 3,3or the spiked disk F. Remark 1.5. Alternative proofs of Corollaries 1.3 and 1.4 can be found in [1] among other characterizations of (properly) planar polyhedra. 2. Basic definitions and proofs of Theorems 1.1 and 1.2. We recall that a Peano continuum Xis a metrizable compact connected locally connected space. When compactness is replaced by local compactness the space Xis called a Peano generalized continuum. Any Peano generalized continuum is arc connected by ([9]; 4.2.5). Moreover, it follows from ([4]; 4.4.F(c)) that any Peano generalized continuum is separable and hence second countable and σ-compact (([4]; 4.1.16) and ([4]; 3.8.C(b))). The local compactness together with the σ-compactness yield that Xis a countable union S∞ n=1 Knof compact subsets Kn⊆Xwith Kn⊆int Kn+1. Actually, we can assume without loss of generality that each Knis connected and all the components of X−Knare unbounded. Indeed, each Knis contained in a finite union of open connected subsets of compact closure K0 n. If K0 n is not connected we can consider a new K0 nby adding to K0 ncompact and connected subspaces joining its (finite) components. If some components of X−Knare bounded then we consider a new K00 nby adding to Knall the bounded components in X−Kn. The sequence {Kn}n≥1with these two properties will be called a decomposition of X. Given a decomposition {Kn}n≥1of XaFreudenthal end of Xis a sequence ε= (Cn)n≥1of components Cn⊆X−Knwith Cn+1 ⊆Cn. Let F(X) 178 R. AYALA ET AL. be the set of Freudenthal ends of Xand let e(X) denote the cardinal number of F(X). The set b X=X∪F(X) admits a compact topology whose basis consists of the open sets of Xtogether with the sets b Cn=Cn∪{ε∈ F(X) : Cn appears in ε}(n≥1). This topology (which does not depend on the sequence {Kn}n≥1) is called the Freudenthal topology and b Xis called the Freudenthal compactification of X. Moreover, the subspace F(X) turns out to be homeomorphic to a closed subset of the Cantor set (see [5] for details). Aproper map f:X→Yis a continuous map such that f−1(K) is compact for any compact subset K⊆Y. If Xand Yare Peano generalized continua the proper map fis necessarily closed. Moreover, the properness of fis equivalent to the continuity of the extension f+:X+→Y+with f+(∞) = ∞between the corresponding one-point compactifications. In fact, fextends to a continuous map b f:b X→b Ywhich restricts to a continuous map f∗:F(X)→ F(Y). Namely if ε= (Cn)n≥1,then b f(ε) = f∗(ε) = (Dk)k≥1where f(Cnk)⊂Dkfor some increasing subsequence (Cnk)k≥1of ε. A proper map r: [0,∞)→Xis called a ray, and ε∈ F(X) is said to be the end induced by rif r∗(∞) = ε. In fact one can find for any end εa ray r: [0,∞)→Xwhich induces ε. Moreover, two rays induce the same end if and only if they can be joined outside any compact subset K⊆X. In order to apply Claytor’s Theorem [2] in the proofs of Theorems 1.1 and 1.2 we shall use the following lemma Lemma 2.1. Let Xbe a Peano generalized continuum. Then its Freudenthal compactification b Xas well as its one-point compactification X+are Peano continua. P r o o f. As a consequence of the Hahn–Mazurkiewicz Theorem ([9]; 4.2.7) continuous images of Peano continua are Peano continua. Hence the map q:b X→X+with q(F(X)) = ∞shows that it will suffice to prove that b Xis a Peano continuum. If X=S∞ n=1 Knis a decomposition of X, the compactness of Kn+1 implies that the number of components of X−Knis finite. Hence the open sets b Cnform a countable neighbourhood basis of all ε∈ F(X) in b X. As Xis second countable, it follows that b Xis second countable. Therefore Urysohn’s Metrization Theorem ([4]; 4.2.8) yields that b Xis a metrizable space. Moreover, b Xis connected since Xis ([4]; 6.1.11). Finally, a ray r: [0,∞)→Cncan be regarded as an arc b r: [0,∞]→b Cnwith b r(∞)∈ F(X). Hence the open neighbourhoods b Cn⊆b Xare arc connected and so b Xis locally (arc) connected. This finishes the proof. Now we are ready to prove Theorem 1.1. Proof of 1.1. Only (3)⇒(4) needs to be checked. Assume that (4) does not hold. Then by Lemma 2.1 we can apply Claytor’s Theorem to find a subspace PLANARITY OF CONTINUA 179 A⊆b Xhomeomorphic to one of the spaces in S={K5, K3,3, L1, L2}. Since we assume (3), one necessarily has A∩F(X)6=∅. We now proceed to replace Aby a new A0still homeomorphic to a space in Swith A0∩ F(X) = ∅, that is, A0⊆X, which will give a contradiction. Case 1: A=K3,3or K5and no vertex of Ais in F(X). In order to obtain A0from Awe firstly observe that for each edge E⊆Athe intersection F=E∩F(X) is homeomorphic to a closed subset of the middle-third Cantor set. Hence we can cover Fby finitely many disjoint open sets W1, . . . , Wkin b Xsuch that all W0 i=Wi∩Xare connected components of the complement X−Kof a certain compact set K⊆X(depending on F). Moreover, we can assume without loss of generality that Wi∩E0=∅for any edge E06=E of A. Furthermore, the intersections Fi=F∩Wiare also closed in F, and hence compact subsets of E. Let xiand yidenote the first and last element in Firespectively. Here we identify Ewith [0,1] by a linear homeomorphism. Notice that xiand yiare not vertices of A. Let x0 i≤xiand y0 i≥yibe points in Fi. Then we replace the segment < x0 i, y0 i>⊆Eby an arc Ci⊆W0 i joining x0 ito y0 i. By proceeding in this way for each edge of Awe obtain a new graph A0⊆b Xwhich is homeomorphic to A. We have a contradiction, and hence Case 1 is finished. Case 2: A=K3,3or K5and some vertex of Ais in F(X). By definition of the Freudenthal topology of b Xwe can find a basis of open neighbourhoods {Un}of vin b Xsuch that U0 n=Un− F(X)⊆Xis arc connected. Moreover, we can assume that Uncontains no vertex other than v. Since F(X) is 0-dimensional, any edge incident to vmeets U0 n. We now take an arc γ1 in U0 njoining two of the edges containing v, say Γ1and Γ2. If D1⊆Γ1 is the open segment from vto q1=γ1∩Γ1we consider the new graph A1= (A−D1)∪γ1. Let U2⊂U1be a new neighbourhood of vin b Xwith γ1∩U2=∅. Let γ2⊆U0 2be an arc joining Γ2⊆A1to another edge of A other than Γ1, say Γ3. Notice that Γ3⊆A1. Let A2= (A1−D2)∪γ2where D2⊆Γ2is the open segment from vto q2=γ2∩Γ2. If A=K3,3we finish here, otherwise we still have to take a new neighbourhood U3⊆U2of vin b X. We can proceed in this way for each vertex in A∩ F(X), and we get a new graph A0⊆b Xfor which A0∩F(X) does not contain vertices of A0and so we are in Case 1. Notice that A0is homeomorphic to Aif A=K3,3and it contains a subgraph homeomorphic to K3,3if A=K5. See Fig. 4. A=K5 v sBBBBB B       @@@@@      A0 q1 q2 q3 γ1 γ2 γ3 Fig. 4 180 R. AYALA ET AL. Case 3: A=L1or L2. Let Ni=Li−Σibe the complement of the segment Σiin Li. If A∩ F(X) contains the limit point pi=Σi∩Ni, the connectedness of Ximplies the existence of an arc γ⊆Xfrom the segment Σito Ni. Here we use the 0-dimensionality of F(X) to ensure Ni∩X6=∅ and Σi−{pi}∩X6=∅. Then the union A∪γcontains a copy of K5if A=L2 or a copy of K3,3if A=L1. That is, we are in the previous cases. If pi6∈ A∩ F(X) then we can find a neighbourhood Ω of piin Xwith Ω∩ F(X) = ∅. Moreover, Ω ∩A⊆Xcontains a homeomorphic copy of Li and hence we reach a contradiction with assumption (3). The proof is now complete. Remark 2.2. The equivalences (1)⇔(2)⇔(3) correspond to the extension of Dirac–Schuster Theorem [3] to Peano generalized continua. We now proceed to prove Theorem 1.2. Proof of 1.2. We shall show (a)⇒(c)⇒(b)⇒(a). In fact, only (c)⇒(b) needs to be checked. We use the notation of the proof of 1.1. Assume on the contrary that X+is not embeddable in S2. According to 2.1 and Claytor’s Theorem we can find a closed subspace A⊆X+homeomorphic to a continuum in the family {K5, K3,3, L1, L2}. Moreover, we can assume ∞ ∈ Asince otherwise A⊆Xand this contradicts (c). If A=K5or K3,3then it easily follows that A− {∞} is one of the Halin graphs in Fig. 4. In case A=Li, we can also assume pi=∞since otherwise we can always get a copy of Liin any neighbourhood Ω of piwith ∞ 6∈ Ω. So assume pi=∞. Since Xis connected we can find an arc γ⊆Xjoining the segment Σi− {pi}to Ni=Li−Σi. Then (A− {pi})∪γcontains a copy of L∞ 5if A=L2or L∞ 3,3if A=L1. This contradicts (c) and the proof is finished. Remark 2.3. Notice that the one-point compactification of a Halin graph is a Kuratowski graph. Moreover, if we join two vertices of different infinite edges in K∞ 5we get a copy of K∞ 3,3embedded in the new graph. When we proceed in the same way with K∞ 3,3we get an embedding of L∞ 3,3. And, if we join three vertices of three different infinite edges of K∞ 5we get an embedding of L∞ 3,3in the new graph. Corollary 2.4. Let Pa planar Peano generalized continuum with e(P) =k. Then Pis not properly planar if and only if it contains a Halin graph Hwith e(H)≤k. P r o o f. If H⊆Pis a Halin graph with e(H)> k then at least two ends of Hare the same in P. Since Pis planar, Remark 2.3 yields a new Halin graph H0with e(H0)< e(H). The result follows after at most three steps. PLANARITY OF CONTINUA 181 Corollary 2.5. Let Pbe a one-ended Peano generalized continuum. Then Pis planar if and only if Pis properly planar. Remark 2.6. The equivalence (a)⇔(c) in 1.2 is the extension to Peano generalized continua of the characterization of Halin and Thomassen of properly planar locally finite connected graphs ([6] and ([10]; Cor 4.1)). In fact, the proof of 1.2 shows that Claytor’s Theorem implies this characterization since Li(i= 1,2) cannot be embedded in any graph H(otherwise the points of valence ≥3 define a set of vertices in Hhaving pias a cluster point in the topology of H). Acknowledgements. This work was partially supported by the project DGICYT PB96-1374. The authors thank the referee for his/her helpful suggestions. REFERENCES [1] R. Ayala, A. M´arquez and A. Quintero, On the planarity of infinite 2-complexes, Abh. Math. Sem. Hamburg 67 (1997), 137–148. [2] S. Claytor, Peanian continua not imbeddable in a spherical surface, Ann. of Math. 38 (1937), 631–646. [3] G. Dirac and S. Schuster, A theorem of Kuratowski, Indag. Math. 16 (1954), 343– 348. [4] R. Engelking, General Topology, Heldermann, 1989. [5] H. Freudenthal, ¨ Uber die topologischer Ra¨ume und Gruppen, Math. Z. 33 (1931), 692–713. [6] R. Halin, Zur h¨aufungspunktfreien Darstellung abz¨ahlbarer Graphen in der Ebene, Arch. Math. (Basel) 17 (1966), 239–243. [7] K. Kuratowski, Sur le probl`eme des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283. [8] S. Mardeˇsi´c and J. Segal, A note on polyhedra embeddable in the plane, Duke Math. J. 33 (1966), 633–638. [9] A. W. Schurle, Topics in Topology, North-Holland, 1979. [10] C. Thomassen, Straightline representations of infinite planar graphs, J. London Math. Soc. 16 (1977), 411–423. R. Ayala and A. Quintero Departamento de Geometr´ıa y Topolog´ıa Facultad de Matem´aticas Universidad de Sevilla Apartado 1160 41080-Sevilla, Spain E-mail: quin[email protected] M. J. Ch´avez Departamento de Matem´atica Aplicada I Escuela Universitaria de Arquitectura T´ecnica Universidad de Sevilla Avda. Reina Mercedes s/n 41012-Sevilla Spain E-mail: mjchav[email protected] Received 6 May 1996; revised 15 April 1997