Houston Journal of Mathematics. c 2013 University of Houston Volume 39, No. 2, Communicated by Charles Hagopian

Transcription

1 Houston Journal of Mathematics c 2013 University of Houston Volume 39, No. 2, 2013 DENDRITES WITH A COUNTABLE SET OF END POINTS AND UNIVERSALITY W LODZIMIERZ J. CHARATONIK, EVAN P. WRIGHT, AND SOPHIA S. ZAFIRIDOU Communicated by Charles Hagopian Abstract. We introduce a notion of ramification degree for dendrites and we use it to show that in the family of all dendrites with a countable set of end points, there is no universal element. Moreover, we characterize some classes of dendrites defined using this ramification degree. Finally, we investigate the problem of existence of minimal dendrite in some families of dendrites with a countable set of end points. 1. Introduction A space Z is said to be universal in a class F of spaces provided that Z F and for each X F there exists an embedding h : X Z. We recall some results concerning the existence of universal element in the families of dendrites. 1. There exists a universal element in the family of all dendrites ([13]). 2. For each n {3, 4,... }, there is a universal element in the family of dendrites with orders of points less than or equal to n ([8], Chapter X, 6, p. 322). 3. There exists a universal element in the family of all dendrites with a closed set of end points ([2]). 4. In the family of all dendrites with a closed, countable set of end points there is no universal element ([14]). In this paper, we prove that in the family of all dendrites with a countable set of end points, there is no universal element Mathematics Subject Classification. 54C25, 54F50, 54G12. Key words and phrases. Dendrite, scattered set, minimal space, universal space, ramification degree. 651

2 652 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU 2. Preliminaries All spaces under consideration are metrizable and separable. By a continuum, we mean a nonempty, compact, and connected space. A dendrite is a locally connected continuum containing no simple closed curve. It is known that any dendrite has a basis of open sets with finite boundaries, and therefore is hereditarily locally connected ([7], 51, VI, Theorem 4, p. 301 and IV, Theorem 2, p. 283). Hence every subcontinuum of a dendrite is a dendrite. The order of point x in a space X, written ord(x, X), is the least cardinal or ordinal number κ such that x has arbitrarily small neighborhood in X with boundary of cardinality κ ([7], 51, I, p. 274). A point x is of order ω in X provided that x has arbitrarily small neighborhood in X with finite boundary but ord(x, X) n for every natural number n. For a point x of a dendrite X, the number of components of X \ {x} is equal to ord(x, X) whenever either of them is finite ([12], (1.1), (iv), p. 88). In the case that ord(x, X) = ω, the components of X \ {x} form a null sequence ([12], (2.6), p. 92). Points of order one are called end points and points of order 3 are called ramification points. A tree is a dendrite with a finite number of end points. We denote by E 2 a plane with an orthogonal coordinate system. Given a, b E 2, the straight line segment joining a and b is denoted by ab. Given a subset M of a space X, we denote by M or cl X M the closure and by M d the set of all limit points of M in X. Let X be a dendrite. Given a, b X, we denote by ab the unique arc from a to b in X, and by (ab) the set ab \ {a, b}. Also, we denote by E(X) and R(X) the set of all end points and of all ramification points of X, respectively. The set of all limit points of E(X) is denoted by E d (X). The set of all limit points of R(X) is denoted by R d (X). We also denote by R ω (X) the set of all points of order ω of X. The following facts are well known (see [3], page 10, Propositions , and [10], Lemma 4, p. 426). Fact 2.1. For any dendrite X, we have E d (X) = R d (X) R ω (X). Fact 2.2. If Y and X are dendrites and Y X, then (a) R(Y ) R(X), (b) E d (Y ) E d (X), and (c) card(e(y )) card(e(x)).

3 DENDRITES WITH A COUNTABLE SET OF END POINTS 653 For every ordinal α, the α-derivative of a space M is defined by induction as follows: M (0) = M, M (α+1) is the set of all limit points of M (α) in M (α), and M (α) = β<α M (β) for a limit ordinal α ([6], 24, IV). If M (α) = for some ordinal α, then the least such ordinal is called the type of M and is denoted by type(m). Note that if M is a subset of a space X, then M (1) = M M d, and the set of all isolated points of M in M coincides with the set M \ M d. A space M is said to be scattered provided that every non-empty subspace of M has an isolated point. Let ω 1 denote the first uncountable ordinal. For each isolated ordinal α > 0 we will denote by α 1 the unique ordinal β such that α = β + 1. The notation α 1 will be applied only to non-limit ordinal α. It is not difficult to obtain the following facts: Fact 2.3. Any scattered space is countable ([6], 23,V). Fact 2.4. A space M is scattered if and only if there exists an ordinal α < ω 1 such that type(m) = α. Fact 2.5. A compact space M is countable if and only if M is scattered. Fact 2.6. If a space M is scattered and M M, then type(m) type( M). Fact 2.7. If a space M is compact and type(m) = α, then the ordinal α is isolated and the set M (α 1) is finite. We recall that the first point map r : X Y for dendrite Y contained in a dendrite X is defined by letting r(x) = x if x Y, and otherwise letting r(x) be the unique point r(x) Y such that r(x) is a point of any arc in X from x to any point of Y (see [9], 10.26, p. 176). Note that the first point map is monotone. 3. Some examples Let I denote the segment [0, 1], C denote the Cantor ternary set in I, and G denote the Gehman dendrite (see Section 5 for detailed definition of G). Example 3.1. For any ordinal α < ω 1, there exists a dendrite G α such that the set E(G α ) is scattered and closed, and type(e(g α )) = α + 1. For α = 0, we can choose any segment. For 0 < α < ω 1, the dendrite G α is defined as the subcontinuum of the Gehman dendrite G which is irreducible with respect to containing a certain set E α C {0} with E α (α) = {(0, 0)} (see [3], page 21).

4 654 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU We quote the definition of E α for the reader s convenience. We define E 1 = ({0} {1/3 n : n {0, 1, 2,...}}) {0} C {0}. Let β be an ordinal such that 1 < β < ω 1, and suppose that the sets E α have been defined for all α < β. We associate to β a sequence {α n } n=0 of ordinals less than β as follows: (i) if β = α + 1, then α n = α for all n, and (ii) if β is a limit ordinal, then β = lim α n. For each n {0, 1, 2,...}, we locate in [2/3 n+1, 1/3 n ] {0} C {0} a copy E n α n of E αn diminished 3 n+1 times in such a way that (E n α n ) (αn) = (2/3 n+1, 0), and define E β = n=0 En α n. Example 3.2. For any ordinal α such that 0 < α < ω 1, there exists a dendrite J α such that the set E(J α ) is scattered but not closed, and type(e(j α )) = α. Set J α = G α pq, where G α is the dendrite defined in Example 3.1, and q = ( 1, 0) and p = (0, 0) are points of E 2. In contrast to Fact 2.6 we have the following example. Example 3.3. For any ordinals β and α such that 0 < β < α < ω 1, there exist dendrites X β and J α X β such that type(e(x β )) = β and type(e(j α )) = α. Let J α be the dendrite defined in Example 3.2, and let E (β) (J α ) = {x 1, x 2,... }. Consider a family of disjoint arcs {x i e i } i=1 such that lim diam(x ie i ) = 0 and i x i e i J α = {x i } for each i. Set X β = J α x i e i. It is easy to see that X β is a dendrite and E(X β ) = i=1 {e 1, e 2,...} (E(J α ) \ E (β) (J α )). Since each e i is an isolated end point of X β and type(e(j α ) \ E (β) (J α )) = β, we conclude that type(e(x β )) = β. Observe that for the above-defined dendrites G α, J α, and X α with scattered sets of end points, the sets E d (G α ), E d (J α ), and E d (X α ) are also scattered. However, we also have the following examples. Example 3.4. There exists a dendrite D C with a countable set of end points such that E(D C ) is scattered and E d (D C ) = C. Define D C as the union of [0, 1] and of countably many vertical segments emanating from the midpoints of contiguous intervals to C, whose lengths are equal to the lengths of the corresponding intervals. Example 3.5. There exists a dendrite D I with a countable set of end points such that E(D I ) is scattered and E d (D I ) = I.

5 DENDRITES WITH A COUNTABLE SET OF END POINTS 655 Define D I as the union of I = [0, 1] and of countably many vertical segments emanating from the points of the form 2k 1 2 for n {1, 2,... } and k n {1,..., 2 n 1 }, where the length of the segment at 2k 1 2 is equal to 1 n 2. n 4. Ramification degree In this section we present the definition and some basic properties of ramification degree. Let X be a continuum and let S X. By irr(s) we denote the subcontinuum of X which is irreducible about S. It is known that any continuum contains an irreducible subcontinuum about any of its non-empty subsets ([12], (11.2), p. 17). Definition 1. Let X be a dendrite. For each ordinal α < ω 1, we define by induction a subcontinuum X (α) of X as follows: X (0) = X, X (α+1) = { irr(r(x(α) )), if R(X (α) ) ;, if X (α) = or R(X (α) ) =. X (α) = β<α X (β) if α is a limit ordinal. Obviously each X (α) is either a subcontinuum of X or is empty. Observe that if α < β, then X (α) X (β) : X (0) X (1) X (2) X (α)..., α < ω 1 We recall the following known theorem (see [1], Theorem 31, p. 162 or [6], 24, II, Theorem 2). Theorem 4.1. If X is a space with a countable basis, then for every well ordered decreasing system of closed subsets of X indexed by all ordinal numbers α < ω 1 F 0 F 1 F 2 F α, α < ω 1 there exists an ordinal α such that F β = F α for each β α: that is, F α = F α+1 = F α+2 =. Corollary 4.2. For every dendrite X, there exists a countable ordinal α such that X (β) = X (α) for each β α: that is, X (α) = X (α+1) = X (α+2) = Definition 2. If X (α) = for some ordinal α, then the least such ordinal α is called the ramification degree of X and is denoted by rdeg(x).

6 656 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU Proposition 4.3. A dendrite X has noisolatedend points if and only if X = X (1). Proof. Let X be a dendrite that has no isolated end points. E d (X) = R d (X) R ω (X) R(X). Hence, Then E(X) X = irr(e(x)) irr(r(x)) = X (1). Conversely, suppose that X = X (1), and that X has an isolated end point e. Then there exists an arc ex in X such that ex R(X) =. It follows that X (1) = irr(r(x)) X \ ex. Hence X (1) X, which is a contradiction. Remark. Since the Gehman dendrite G has no isolated end points (see Section 5 for detailed description of G), it follows that G (α) = G for each ordinal α < ω 1. Hence, G has no ramification degree. On the other hand, for the dendrites defined in Examples we have rdeg(g α ) = rdeg(j α ) = rdeg(x β ) = α + 1 and rdeg(d C ) = rdeg(d I ) = 2. Proposition 4.4. If E(X) \ E d (X) and R(X) for some dendrite X, then for each e E(X) \ E d (X) there exists a point x e X (1) such that: (i) ex e X (1) = {x e }; (ii) (ex e \ {x e }) (ẽxẽ \ {xẽ}) = for ẽ, e E(X) \ E d (X), ẽ e; (iii) X \ X (1) = { ex e \ {x e } : e E(X) \ E d (X) }. Proof. Since E d (X) R(X) X (1) and X (1) does not contain isolated end points of X, E(X) (X \ X (1) ) = E(X) \ E d (X). Let r : X X (1) be the first point map for X (1). For every e E(X) \ E d (X), we set x e = r(e). Then (i) follows from the definition of r. Since R(X) X (1), (ii) follows from the fact that for each isolated end point e of X, we have (ex e \ {x e }) R(X) =. Finally, (iii) follows from the fact that each component of X \ X (1) contains e E(X) (X \ X (1) ). Proposition 4.5. X \ X (α) = β<α (X (β) \ X (β+1) ) for any dendrite X and for any ordinal α < ω 1. Proof. Since X (α) = β α X (β), it follows that for every x X (α) the ordinal β x = min{β α : x X (β) } is uniquely determined. Since x X (β) for each β < β x, the ordinal β x is isolated. Thus x X (βx 1) \ X (βx), and β x 1 < α. Proposition 4.6. If rdeg(x) = α for a dendrite X, then the ordinal α is isolated and the set X (α 1) is either an arc or a single point.

7 DENDRITES WITH A COUNTABLE SET OF END POINTS 657 Proof. On the contrary, suppose that α is a limit ordinal. Consider an increasing sequence of ordinals {β i } i=1 such that lim β i = α. Since each X (βi) is non-empty i and compact, and X (βi) X (βi+1), it follows that i=1 X (β i). Since {X (β) } β<α is a decreasing system of closed subsets indexed by all ordinals β < α, we have i=1 X (β i) = β<α X (β) = X (α). Hence X (α), which is a contradiction. On the other hand, if rdeg(x) = α, then X (α) = and X (α 1). Hence R(X (α 1) ) =. Therefore X (α 1) is either an arc or a single point. Proposition 4.7. If X and Z are dendrites and X Z, then X (α) Z (α) for each α < ω 1. Proof. The proof is by induction on α. We have X (0) = X Z = Z (0). Let α 0 be an ordinal such that 0 < α 0 < ω 1, and suppose that X (α) Z (α) for each α < α 0. Case 1. α 0 = α + 1. Then X (α) Z (α) by induction. If either X (α) = or R(X (α) ) =, then X (α0) = Z (α0). If R(X (α) ), then R(X (α) ) R(Z (α) ). Thus X (α0) = irr(r(x (α) )) irr(r(z (α) )) = Z (α0). Case 2. α 0 is a limit ordinal. Then since X (α) Z (α) for each α < α 0, we obtain X (α0) = α<α 0 X (α) α<α 0 Z (α) = Z (α0). Corollary 4.8. If X and Z are dendrites, X Z, and Z has a ramification degree, then rdeg(x) rdeg(z). Theorem 4.9. If f : X Y is a monotone map of a dendrite X onto a dendrite Y, then Y (α) f(x (α) ) for each ordinal α. Proof. We first prove that the restriction of f to each subcontinuum of X is monotone. Indeed, let S be a subcontinuum of X and y f(s). Denote g = f S. Clearly, g 1 (y) = f 1 (y) S. Since the intersection of two connected subsets of dendrite is connected (see [9], Theorem 10.10), the set g 1 (y) is connected. Now we prove the Theorem by induction on α. The case α = 0 is trivial (we have Y (0) = Y = f(x) = f(x (0) )). Suppose that Y (α) f(x (α) ) for each α < α 0. If α 0 is isolated, then, by induction hypothesis, Y (α0 1) f(x (α0 1)). Since f : X X(α0 1) (α 0 1) f(x (α0 1)) is monotone, f X(α0 is weakly confluent. It 1) follows that (see [4], Theorem II.1) R(f(X (α0 1))) f(r(x (α0 1))).

8 658 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU Hence, R(Y (α0 1)) R(f(X (α0 1))) f(r(x (α0 1))) f(x (α0 )). Since f(x (α0 )) is a continuum containing the set R(Y (α0 1)), we have Y (α0 ) = irr(r(y (α0 1))) f(x (α0 )). If α 0 is a limit ordinal, then by hypothesis, Y (α) f(x (α) ) for each α < α 0. So, from compactness of X (α) for each α < α 0 and continuity of f, it follows that Y (α0 ) = Y (α) f(x (α) ) = f = f(x (α0 )). α<α 0 α<α 0 α<α 0 X (α) Corollary If f : X Y is a monotone map of a dendrite X onto a dendrite Y, then rdeg(y ) rdeg(x). Note that Corollary 4.10 is in fact a generalization of Corollary 4.8, since the first point map as defined after Fact 2.7 is monotone. 5. Dendrites with a countable set of end points Now we will show how ramification degree can be applied to prove nonexistence of universal elements in some classes of dendrites. Notations. We set L 0 = { }, and for n {1, 2,... } we denote by L n the set of all ordered n-tuples i 1... i n, where i t = 0 or i t = 1. We also set L = n=0 L n. For ī = i 1... i n L n, n 1, we denote by ī0 the (n + 1)-tuple i 1... i n 0 and by ī1 the tuple i 1... i n 1. For ī L 0, we put ī0 = 0 and ī1 = 1. Let us consider the Cantor set C. We set C = C and denote by C ī, where ī = i 1... i n L n and n 1, the set of all points of C for which the t th digit in the ternary expansion, t {1,..., n}, coincides with 0 if i t = 0 and with 2 if i t = 1. For every c C, we denote by ī(c, n) the uniquely determined ī L n such that c C ī. Obviously, {c} = n=0 Cī(c,n). For ī L, we set a ī = (min{x x C ī }, 0) and b ī = (max{x x C ī }, 0). Note that C ī {0} = a ī b ī (C {0}). Gehman dendrite. We quote the construction of the Gehman dendrite (see [10]). Let l n, for n {0, 1,... }, be the line y = (2 3 n ) 1 of E 2. Put v = ( 1 2, 1 2 ), v 0 = v a l 1, v 1 = v b l 1, V 1 = {v 0, v 1 }, and H 0 = v v 0 v v 1.

9 DENDRITES WITH A COUNTABLE SET OF END POINTS 659 v v 0 v 1 l 1 v 00 v 01 v 10 v 11 l 2 l 3 Figure 1. Tree H 2 Suppose that, for k {1,..., n}, the sets V k = {v ī : ī L k } l k and the trees H k 1 has been defined. For each ī L n, we define v ī0 = v ī a ī l n+1 and v ī1 = v ī b ī l n+1. Put V n+1 = {v ī : ī L n+1 }, and ( ) H n = H n 1 {vī v ī0 v ī v ī1 : ī L n }. The tree H 2 is pictured in Figure 1. The set G ω = n=0 H n is called the standard zero-one-tree and the set G = cl(g ω ) is called the Gehman dendrite. The following facts will be used in Section 6. Fact 5.1. If H n is the tree constructed above, then rdeg(h n ) = n + 1. Fact 5.2. For each n {1, 2,... }, the tree H n+1 is the union of an arc and two copies of H n attached to the ends of the arc. Gehman dendroid. ([11], page 205 or [10], page 423.) Dendroid is the hereditarily unicoherent and arcwise connected continuum. A space S is called a zero-one-tree provided that there exists a continuous, one-to-one, ( and onto map f : G ω S such that for every c C, the set ) f v v ī(c,n) is a half-open arc and is closed in S. n=1 A dendroid X that contains a zero-one-tree as a dense subset is called a Gehman dendroid. Lemma 5.3. Let X be a dendrite and let M E(X) be such that M (1). For each e M (1) and each arc ep X, there exist sequences {e n } n=1 M and {r n } n=1 (ep) R(X) such that (ep) e n r n = {r n } and r n+1 (er n ) for all n.

10 660 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU Proof. Let r : X ep be the first point map for ep. Since e M (1), e = lim e i, where e i M. Since r is continuous (see [9], Lemma 10.25, page 176), it follows that e = r(e) = lim r(e i ). It is clear that {r(e i )} i=1 R(X) (ep], and that there exists a subsequence {r(e in )} n=1 of {r(e i )} i=1 such that r(ein+1 ) (er(e in )). Set e n = e in and r n = r(e n ). Since r n is the unique point of ep such that r n is a point of any arc in X from e n to any point of ep, we conclude that e n r n (ep) = {r n }. Theorem 5.4. If the set of end points of dendrite X is not scattered, then X contains a copy of the Gehman dendrite. Proof. We shall prove that there is an embedding h : G ω X. Then cl X (h(g ω )) is a Gehman dendroid and therefore topologically contains the Gehman dendrite (see [11], Theorem 8 in page 212). Since E(X) is not scattered, there exists M E(X) such that M (1) = M. Obviously, M is infinite. Let e 0, e1 M = M (1). From Lemma 5.3, it follows that there exists r (e 0 e1 ) R(X). Since e0, e1 M (1), from Lemma 5.3 it follows that there exist points r 0 (r e 0 ) R(X), r 1 (r e 1 ) R(X), and e 1 0, e 0 1 M \ {e 0, e1 } such that r 0 e 1 0 r e 0 = {r 0} and r 1 e 0 1 r e 1 = {r 1}. Put H 0 = r r 0 r r 1 and R 1 = {r 0, r 1 }. Also put e 0 0 = e 0, e1 1 = e 1, and E 1 = {e 0 ī, e1 ī : ī L 1 }. Let n 1 be an integer and suppose that for each integer k {1,..., n} we have defined: a tree H k 1, a set of distinct points R k = {r ī : ī L k } R(X), and a set of distinct points E k = {e 0 ī, e1 ī : ī L k } M such that : (i) E(H k 1 ) = R k ; (ii) H k 1 E k = ; (iii) (e 0 ī e1 ī ) H k 1 = {r ī } for each ī L k ; (iv) e 0 ī e1 ī e0 j e1 j = for all ī, j L k with ī j. Let ī L n. Since e 0 ī, e1 ī M (1) and r ī (e 0 ī e1 ), from Lemma 5.3 it follows that ī there exist points r ī0 (r ī e 0 ī ) R(X), rī1 (r ī e 1 ī ) R(X), and e1 ī0, e0 ī1 M \ E n such that r ī0 e 1 ī0 rī e 0 ī = {rī0 } and r ī1 e 0 ī1 rī e 1 ī = {rī1 }. Put e 0 ī0 = e0 ī and e1 ī1 = e1 ī. Set R n+1 = {r ī : ī L n+1 }, E n+1 = {e 0 ī, e1 ī : ī L n+1 }, and H n = H n 1 ( {r ī r ī0 r ī r ī1 : ī L n }).

11 DENDRITES WITH A COUNTABLE SET OF END POINTS 661 The sets H n, R n+1, and E n+1 satisfy the properties (i) (iv) for k = n + 1. Put H ω = (r ī r ī0 r ī r ī1 ). H n = n=0 ī L Now consider the standard zero-one-tree G ω = (v ī v ī0 v ī v ī1 ). For each ī L, let h ī : v ī v ī0 v ī v ī1 r ī r ī0 r ī r ī1 be a homeomorphism such that h ī (v ī ) = r ī, h ī (v ī0 ) = r ī0, and h ī (v ī1 ) = r ī1. For every ī L, put V ī = v ī v ī0 v ī v ī1 and Λ ī = r ī r ī0 r ī r ī1. We define h : G ω H ω by letting h(x) = h ī (x) for x V ī. It is easily seen that h is well defined, one-to-one, and onto. Since the family {V ī : ī L} is a locally finite cover of G ω by (closed) arcs such that for each ī L the restriction h Vī = h ī is continuous, and h ī = h j for all ī, Vī V j Vī j L, h is continuous V j (see [5], Proposition , p. 100). Similarly, h 1 : H ω G ω is continuous. Thus h is a homeomorphism. Theorem 5.5. The following conditions are equivalent for a dendrite X: (i) E(X) is countable; (ii) E(X) is scattered subspace of X; (iii) rdeg(x) = α for some ordinal α < ω 1. Proof. (i) (ii) Let the set E(X) be countable, and suppose that it is not scattered. Then by Theorem 5.4, X contains a copy of the Gehman dendrite. From Fact 2.2(c) it follows that card(e(g)) card(e(x)). Thus E(X) is uncountable, which is a contradiction. (ii) (iii) Let E(X) be scattered. By Corollary 15, there exists a least ordinal α such that X (α) = X (β) for each β α. It suffices to show that X (α) =, since then rdeg(x) = α. On the contrary, suppose that X (α). Since X (α) = X (α+1), by Proposition 4.3 the dendrite X (α) has no isolated end points. Thus E(X (α) ) is not scattered. By Theorem 5.4, the dendrite X (α) contains a copy of the Gehman dendrite. Hence X contains a copy of the Gehman dendrite. Then from Fact 2.2(c), it follows that E(X) is uncountable. Consequently, from Fact 2.3 it follows that the subspace E(X) is not scattered, which is a contradiction. (iii) (i) From Proposition 4.6, we have that α = α for some α 0, and that X (α0) is either a point or an arc. Also, X \ X (α0) = β<α 0 (X (β) \ X (β+1) ) by Proposition 4.5. Let β < α 0. Since X (β+1), from the definition of X (β+1) it follows that R(X (β) ). Since X (β) X (β+1), from Proposition 4.3 it follows that the set of isolated end points of X (β) is not empty. So, by Proposition 4.4, we have ī L

12 662 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU X (β) \ X (β+1) = { ex e \ {x e } : e E(X (β) ) \ E d (X (β) ) }, where: (i) ex e X (β) = {x e }; (ii) (ex e \ {x e }) (ẽxẽ \ {xẽ}) = for ẽ, e E(X (β) ) \ E d (X (β) ), ẽ e. It is easily seen that E(X) \ X (α0) ( β<α E(X(β) 0 ) \ E d (X (β) ) ). Since the set of all isolated end points of any dendrite is at most countable, we have that the set E(X (β) ) \ E d (X (β) ) is countable for each β < α 0. On the other hand, E(X (α0)) consists of at most two points. Hence, E(X) is countable. Theorem 5.6. If T is a family of dendrites with a countable set of end points, such that for each α < ω 1 there exists T α T with rdeg(t α ) > α, then there is no dendrite with a countable set of end points topologically containing every element of T. Proof. On the contrary, suppose that such a dendrite Z exists. Then, from Theorem 5.5, rdeg(z) = α Z for some ordinal α Z < ω 1. By Corollary 4.8, for each dendrite X Z we have rdeg(x) α Z. Hence, the dendrite T αz with a ramification degree greater than α Z can not be embedded into Z, which is a contradiction. Let W denote the union of the segment [ 1, 1] and countably many vertical segments emanating from the points of the form 1/n, n {1, 2,... }, where the length of the segment at 1/n is 1/n. It is proved in [2] that there exists a universal element in the family of all dendrites containing no copy of W. Observe that the dendrites G α defined in Example 3.1 for all α < ω 1 each have a closed and countable set of end points and contain no copy of W. Moreover, rdeg(g α ) = α + 1, so we obtain the following results. Corollary 5.7. In the family of all dendrites with a countable set of end points, there is no universal element. Corollary 5.8. In the family of all dendrites with a countable set of end points containing no copy of W, there is no universal element. Corollary 5.9. There is no dendrite with a countable set of end points which topologically contains every dendrite with a closed and countable set end points. Corollary There is no dendrite with a countable set of end points which topologically contains every dendrite X with a countable set of end points such that X contains no copy of W.

13 DENDRITES WITH A COUNTABLE SET OF END POINTS 663 Figure 2. Dendrite Y 3 6. Minimal dendrites In this section we will investigate the existence of minimal dendrites in some families defined using ramification degree. A space Z is said to be minimal in a class F of spaces provided that Z F, and for each X F there exists an embedding h : Z X. In the sequel, every time we talk about attaching one dendrite to another we mean attaching by the distinguished points. For n {0, 1, 2,... }, the symbol H n denotes the tree as in the construction of the Gehman dendrite just before Fact 5.1. The distinguished point of H n is the vertex v. Moreover, define Y 0 to be a single point, Y 1 to be the simple triod, and Y n for n 2 to be the simple triod with a copy of H n 2 attached to each end point. The dendrite Y 3 is pictured in Figure 2. The facts established in the following propositions are consequences of the construction. Their proofs are left to the reader. Proposition 6.1. For every number n {0, 1, 2,... }, the following hold: (1) rdeg(h n ) = n + 1; (2) rdeg(y n ) = n + 1; (3) (H n ) (n) is an arc; (4) (Y n ) (n) is a point; (5) every ramification point of H n is of order 3; (6) every ramification point of Y n is of order 3. Theorem 6.2. For a dendrite X having a ramification degree and a number n {0, 1, 2,...}, the following conditions are equivalent: (1) rdeg(x) n + 1 and X (n) is nondegenerate; (2) X contains a copy of H n.

14 664 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU Proof. First assume (2). Then rdeg(x) n + 1 from Proposition 6.1 (1) and Corollary 4.8. The fact that X (n) is nondegenerate follows from Proposition 4.7 and Proposition 6.1 (3). This concludes the implication (2) (1). We will prove the implication (1) (2) by induction. If n = 0, then X = X (0) is nondegenerate, so it contains an arc (which H 0 is). Suppose that the implication (1) (2) is true for n 1 and that X is a dendrite such that X (n) is nondegenerate. Then there are two different points a and b of R(X (n 1) ). Denote by D and D the components of X \ (ab) that contain a and b respectively. Since a and b are ramification points of X (n 1), we conclude that D (n 1) and D (n 1) are nondegenerate, and thus both D and D contain copies of H n 1. Now the conclusion that X contains a copy of H n follows from Fact 5.2. Corollary 6.3. For each n {0, 1, 2,... }, the dendrite H n is minimal in the family of all dendrites X such that rdeg(x) n + 1 and X (n) is nondegenerate. Theorem 6.4. For a dendrite X having a ramification degree and a number n {0, 1, 2,... }, the following conditions are equivalent: (1) rdeg(x) n + 1; (2) X contains a copy of Y n. Proof. The implication (2) (1) is a consequence of Proposition 6.1 (2) and Corollary 4.8. We will prove the implication (1) (2) by induction. The case n = 0 is trivial. If n = 1, then X (1), so X contains a triod Y 1. Suppose that the implication (1) (2) is true for n 1 and that X is a dendrite such that rdeg(x) n + 1. Then (X (1) ) (n 1) = X (n). Therefore rdeg(x (1) ) n. By induction assumption X (1) contains a copy Ŷn 1 of Y n 1. Note that each e E(Ŷn 1) either belongs to E(X (1) ) or can be join by an arc with a point of E(X (1) ). Since E(X (1) ) R(X), we can suppose without loss of generality that E(Ŷn 1) R(X). For each e E(Ŷn 1) we chose an arc A e X in such a way that: (i) e is an inner point of A e ; (ii) Ŷn 1 A e = {e}; (iii) if e 1, e 2 E(Ỹn 1) and e 1 e 2, then A e1 A e2 =. Clearly, the set Ŷn 1 ( {A e : e E(Ŷn 1)} is a copy of Y n contained in X. Corollary 6.5. For n {0, 1,... }, the dendrite Y n is the minimal dendrite in the family of all dendrites X such that rdeg(x) n + 1.

15 DENDRITES WITH A COUNTABLE SET OF END POINTS 665 In the sequel for each countable ordinal α ω we construct dendrites H α, Y α, and Ỹα such that (a) (H α ) (α) is an arc; (b) (Y α ) (α) is a point; (c) (Ỹα) (α) is a point; (d) there is no dendrite with ramification degree equal to α + 1 that can be embedded into both of the dendrites Y α and Ỹα. We proceed by induction, using the already defined tries H n and Y n. Each countable ordinal α ω can be uniquely expressed as the sum λ + n, where λ is a limit ordinal and n {0, 1, 2,...}. Assume that λ ω is a countable limit ordinal and that the dendrites H α and Y α have been defined for each α < λ. Let λ 1, λ 2,... be a sequence of ordinals less than λ and converging to λ. Let F be the dendrite with only one ramification point and such that the order of the ramification point is ω. Then E(F ) = {e 1, e 2,...}. To construct the dendrite Ỹλ, attach for each n {1, 2,... } a copy of H λn to the point e n of F. In E 2, let P denote the union of the segment [0, 1] {0} and of countably many vertical segments { } [ 1 n 0, 1 n]. To construct the dendrite Yλ, attach copies of H λn to the points ( 1 n, n) 1. The distinguished point of Yλ is either the point p = (0, 0) or the point q = (1, 0). To construct the dendrite H λ, attach copies of Y λ to the end points of H 0 by the distinguished point p = (0, 0). Let n {1, 2,... }. We define: (i) Y λ+n as the union of Y n and a copy of Y λ attached to each end point of Y n by the distinguished point p = (0, 0); (ii) Ỹλ+n as the union of Y n and a copy of Y λ attached to each end points of Y n by the distinguished point q = (1, 0); (iii) H λ+n as the union of H n and a copy of Y λ attached to each end point of H n by the distinguished point p = (0, 0). The distinguished point of H λ+n is the point v of H n. Theorem 6.6. For each α ω, there is no minimal dendrite in the family of all dendrites X such that rdeg(x) α + 1. Proof. On the contrary, suppose that for some countable ordinal α ω there is a minimal dendrite Z in the family of all dendrites X such that rdeg(x) α + 1. Then rdeg(z) α + 1.

16 666 W. J. CHARATONIK, E. P. WRIGHT, AND S. S. ZAFIRIDOU From properties (b) and (c) of dendrites Y α and Ỹα, respectively, it follows that rdeg(y α ) = rdeg(ỹα) = α + 1. Therefore, the minimal dendrite Z can be embedded in both of dendrites Y α and Ỹα. Then, from Corollary 4.8 and property (d) of dendrites Y α and Ỹα, it follows that rdeg(z) < α + 1, which is a contradiction. References [1] P. S. Alexandroff, Introduction to set theory and general topology, Izdat. Nauka, Moskow, 1977 (in Russian). [2] D. Arévalo, W. J. Charatonik, P. P. Covarrubias, and L. Simon, Dendrites with a closed set of end points, Topology Appl. 115 (2001), [3] J. J. Charatonik, W. J. Charatonik, J. R. Prajs, Mapping hierarchy for dendrites, Dissertationes Math. (Rozprawy Mat.) 333 (1994), [4] C. A. Eberhart, J. B. Fugute, G. R. Gorth Jr., Branchpoint covering theorems for confluent and weakly confluent maps, Proc. Amer. Math. Soc. 55(1976), [5] R. Engelking General Topology, Warszawa, [6] K. Kuratowski, Topology, Vol. I, New York, [7] K. Kuratowski, Topology, Vol. II, New York, [8] K. Menger, Kurventheorie, Teubner, [9] S. B. Nadler, Jr. Continuum theory: An introduction, M. Dekker [10] J. Nikiel, A characterization of dendroids with uncountably many end points in the classical sense, Houston J. Math.9 (1983), [11] J. Nikiel On Gehman dendroid, Glasnik Mat. 20(40)(1985) [12] G. T. Whyburn, Analytic Topology, Amer. Math. Soc., [13] T. Ważewski, Sur les courbes de Jordan ne renfermant aucune courbe simple fermée de Jordan, Annales de la Société Polonaise de Mathématique 2 (1923), [14] S. Zafiridou, Universal dendrites for some families of dendrites with a countable set of end points, Topology Appl. 155 (2008), no , Received August 7, 2009 Revised version received January 25, 2011 W lodzimierz J. Charatonik, Missouri University of Science and Technology, Department of Mathematics and Statistics, Rolla, MO, USA address: wjcharat@mst.edu Evan P. Wright, Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA address: evanpw@math.sunysb.edu Sophia S. Zafiridou, DepartmentofMathematics, UniversityofPatras, Patras, Greece address: zafeirid@math.upatras.gr