DENDRITES WITH A COUNTABLE SET OF END

Houston Journal of Mathematics c 2013 University of Houston

Volume 39, No. 2, 2013

DENDRITES WITH A COUNTABLE SET OF END POINTS AND UNIVERSALITY WLODZIMIERZ 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. 2000 Mathematics Subject Classification. 54C25, 54F50, 54G12. Key words and phrases. Dendrite, scattered set, minimal space, universal space, ramification degree. 651

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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) 6= 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 ∈ 2 E , 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 clX 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 Rd (X). We also denote by Rω (X) the set of all points of order ω of X. The following facts are well known (see [3], page 10, Propositions 4.13–4.14, and [10], Lemma 4, p. 426). Fact 2.1. For any dendrite X, we have E d (X) = Rd (X) ∪ Rω (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)).

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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 T M (α) = β 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 < ω1 such that type(M ) = α. Fact 2.5. A compact space M is countable if and only if M is scattered. f is scattered and M ⊆ M f, then type(M ) ≤ type(M f). Fact 2.6. If a space 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 < ω1 , there exists a dendrite Gα such that the set E(Gα ) is scattered and closed, and type(E(Gα )) = α + 1. For α = 0, we can choose any segment. For 0 < α < ω1 , the dendrite Gα is defined as the subcontinuum of the Gehman dendrite G which is irreducible with (α) respect to containing a certain set Eα ⊆ C × {0} with Eα = {(0, 0)} (see [3], page 21).

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We quote the definition of Eα for the reader’s convenience. We define E1 = ({0} ∪ {1/3n : n ∈ {0, 1, 2, ...}}) × {0} ⊆ C × {0}. Let β be an ordinal such that 1 < β < ω1 , and suppose that the sets Eα have been defined for all α < β. We associate to β a sequence {αn }∞ n=0 of ordinals less than β as follows: (i) if β = α + 1, then αn = α for all n, and (ii) if β is a limit ordinal, then β = lim αn . For each n ∈ {0, 1, 2, ...}, we locate in [2/3n+1 , 1/3n ] × {0} ⊆ C × {0} a copy n Eαn of Eαn diminished 3n+1 times in such a way that (Eαnn )(αn ) = (2/3n+1 , 0), S∞ and define Eβ = n=0 Eαnn . Example 3.2. For any ordinal α such that 0 < α < ω1 , there exists a dendrite Jα such that the set E(Jα ) is scattered but not closed, and type(E(Jα )) = α. Set Jα = Gα ∪ pq, where Gα 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 < ω1 , there exist dendrites Xβ and Jα ⊆ Xβ such that type(E(Xβ )) = β and type(E(Jα )) = α. Let Jα be the dendrite defined in Example 3.2, and let E (β) (Jα ) = {x1 , x2 , . . . }. Consider a family of disjoint arcs {xi ei }∞ i=1 such that lim diam(xi ei ) = 0 and i→∞

xi ei ∩ Jα = {xi } for each i. ∞ S Set Xβ = Jα ∪ xi ei . It is easy to see that Xβ is a dendrite and E(Xβ ) = i=1

{e1 , e2 , ...} ∪ (E(Jα ) \ E (β) (Jα )). Since each ei is an isolated end point of Xβ and type(E(Jα ) \ E (β) (Jα )) = β, we conclude that type(E(Xβ )) = β. Observe that for the above-defined dendrites Gα , Jα , and Xα with scattered sets of end points, the sets E d (Gα ), E d (Jα ), and E d (Xα ) are also scattered. However, we also have the following examples. Example 3.4. There exists a dendrite DC with a countable set of end points such that E(DC ) is scattered and E d (DC ) = C. Define DC 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 DI with a countable set of end points such that E(DI ) is scattered and E d (DI ) = I.

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Define DI as the union of I = [0, 1] and of countably many vertical segments emanating from the points of the form 2k−1 for n ∈ {1, 2, . . . } and k ∈ 2n 1 {1, . . . , 2n−1 }, where the length of the segment at 2k−1 2n is equal to 2n . 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 < ω1 , we define by induction a subcontinuum X(α) of X as follows: X(0) = X,  X(α+1) = X(α) =

T

irr(R(X(α) )), ∅,

β