ON A FAMILY OF DENDRITES

We study the open images of members of a countable family of dendrites. We show that only two members of are minimal and only one of them is unique minimal with respect to open mappings. 2000 Mathematics Subject Classification. 54E40, 54F15, 54F50.


Introduction.
Let be a family of topological spaces and F a class of mappings between members of .Then can be quasi-ordered with respect to F, writing for any ( A member X 0 of is said to be • minimal in with respect to F provided that, for each Y in the condition Y ≤ F X 0 implies Y = F X 0 ; • unique minimal in with respect to F provided that for each Y in if Y ≤ F X 0 then Y is homeomorphic to X 0 .Thus, in particular, all spaces in which are homeomorphic to all its images under mappings belonging to F are unique minimal in with respect to F. (See [5, Chapter 3, page 7] for more information.) In this paper, we take as the family Ᏸ of dendrites (i.e., locally connected continua containing no simple closed curves) and as F the class O of open mappings (i.e., ones which map open subsets of the domain onto open subsets of the range).Various properties of the relation ≤ O on the family Ᏸ are discussed in [5,Chapter 6,.Examples of dendrites which are homeomorphic to all its open images can be found, for example, in [2, Corollary, page 493 and the paragraph following it] and in [5,Theorem 6.45,page 30].
Answering a question in [5, Q2(O), page 51] (see also [3,Section 6,2, page 245]) a dendrite C is constructed in [9, Section 2] which is minimal with respect to O and which has two topologically distinct open images, thus is not unique minimal with respect to O (see [9, Proposition 3.5(α)]).The quoted paper contains also a construction of a countable family Ᏺ of dendrites, with C ∈ Ᏺ.Since each member of Ᏺ has a similar structure as the one of C, it is natural to ask about open mapping properties of other members of Ᏺ, especially properties which are related to the minimality of members of Ᏺ with respect to the class O.This is a subject of the present paper.
All spaces considered in the paper are assumed to be metric and all mappings are continuous.A continuum means a compact connected space.Given a space X and its subset S, we denote by cl S the closure of S and by int S its interior in X.As usual N denotes the set of positive integers, and R stands for the space of real numbers.
We will use the notion of order of a point in the sense of Menger-Urysohn (cf.[7, Section 51, I, page 274]), and we denote by ord(p, X) the order of the space X at a point p ∈ X.It is well known (cf.[7, Section 51, pages 274-307]) that the function ord takes its values from the set S = 0, 1, 2,...,n,...,ω,ℵ 0 , 2 ℵ 0 . (1.2) Points of order 1 in a space X are called end points of X; the set of all end points of X is denoted by E(X).Points of order 2 are called ordinary points of X.It is known that the set of all ordinary points is a dense subset of a dendrite.And for each r ∈ {3, 4,...,ω,ℵ 0 , 2 ℵ 0 } points of order r are called ramification points of X; the set of all ramification points is denoted by R(X).It is known that for each dendrite X the set R(X) is at most countable, and that the points of order ℵ 0 and 2 ℵ 0 do not occur in any dendrite.
A space X is said to be universal in a class of spaces if it belongs to the class and it contains a homeomorphic copy of every element of that class.

The construction.
It should be stressed that the construction below is modeled onto the one described in [9, Section 2], and also the proofs of the properties of the dendrites D(r , s) are patterned after the corresponding ones presented in [9, Sections 2 and 3].
To construct the mentioned family Ᏺ of dendrites, we fix some notation and terminology.For n ∈ N let F n denote the simple n-od, that is, a continuum homeomorphic to the cone over a (discrete) set of n points.The vertex of the cone is called the vertex of F n .In the Cartesian coordinates in the plane R 2 put v = (0, 0), and for each n ∈ N let e n = (1/n, 1/n 2 ).Denoting by pq the straight line segment in the plane with end points p and q, define The continua F n and F ω are called fans of order n and ω, respectively.Any fan of order n ∈ N (thus having the set E(F n ) of its end points finite) is also named a finite fan, and F ω is also termed an infinite locally connected fan.Obviously fans F n and F ω are dendrites.An arc pq with end points p and q in a continuum X is called a free arc provided that pq\{p, q} is an open subset of X.If a free arc is not properly contained in another one, it is called a maximal free arc.Then three cases are possible: either both p and q are ramification points (and then it is called an interior free arc), or one of them is a ramification point and the other is an end point of X (and then pq is called an end free arc), or finally both p and q are end points of X.Note that the third possibility holds only in a trivial case when X = pq.Theorem 2.1.For every r ∈ {3, 4,...,ℵ 0 } and s ∈ {0, 1, 2,...,ℵ 0 } there exists a dendrite D(r , s) such that: (2.1.1)each ramification point of D(r , s) belongs to exactly r interior free arcs in D(r , s); (2.1.2) each ramification point of D(r , s) belongs to exactly s end free arcs in D(r , s); (2.1.3)any two ramification points of D(r , s) are contained in an arc in D(r , s) containing only finitely many ramification points of D(r , s).Moreover, conditions (2.1.1),(2.1.2),and (2.1.3)determine the dendrite D(r , s) up to a homeomorphism.
, where v is the common vertex of the two fans.If s = 0, we take F s = {v}, and if s = 1 or s = 2 we understood F 1 s as the union of one or two arcs, respectively, emanating from the point v and disjoint out of this point.Thus X 1 is a fan with the vertex v, either finite or homeomorphic to F ω .In the set E(X 1 ) we distinguish a subset ). Assume that a dendrite X n is defined for some n ∈ N and that in the set E(X n ) of its end points a nonempty subset E n is distinguished.Consider the one point union U = F r −1 ∪ F s where the vertices of the fans F r −1 and F s are identified to a point v(U).Then X n+1 is obtained from X n by attaching to each end point e ∈ E n ⊂ X n a properly diminished copy U(e) = F n+1 r −1 (e) ∪ F n+1 s (e) of U with the points e ∈ X n and v(U(e)) ∈ U(e) identified, in such a way that X n ∩ U(e) = {e}, where F n+1 r −1 (e) and F n+1 s (e) denote the corresponding copies of the fans F r −1 and F s , respectively.Thus X n+1 is a dendrite by its definition.Further, define E n+1 = {E(F r −1 (e)) : e ∈ E n }.
Note that X n ⊂ X n+1 for each n ∈ N. We assume that the diameters of the components of X n+1 \ X n tend to 0 if n tends to infinity.Let f n : X n+1 → X n be a monotone retraction.Thus f n shrinks each of the attached fans U(e) back to its vertex v(U(e)) which is identified with the corresponding end point e ∈ E n ⊂ E(X n ).
Consider the inverse sequence {X n ,f n : n ∈ N} of dendrites X n with monotone bonding mappings f n , and define By [8, Theorem 10.36, page 180 and Theorem 2.10, page 23] the defined inverse limit D(r , s) is a dendrite which is homeomorphic to cl( {X n : n ∈ N}).Neglecting the homeomorphism we can simply write It is evident from the construction that D(r , s) has properties (2.1.1),(2.1.2),and (2.1.3).In [9,Proposition 3.3] it is proved that these properties uniquely determine D(r , s).The proof is then complete.
Statement 2.2.The dendrite D(r , s) is composed exclusively of points of order 1, 2, and r + s, with a convention that, in the case when one of r or s is ℵ 0 , points of order r + s are understood as ones of order ω.
The next statement is a consequence of property (2.1.3).As a consequence of (2.3) and Statement 2.3, we get the following inclusion.
The next inclusion is obvious.
In particular, we have the following: Note that D(ℵ 0 , 1) is C 1 ω of [9].For each integer n ≥ 3, a dendrite G n is constructed in [1, Chapter 4] which is universal in the class of all dendrites with a closed set of end points and of orders of their ramification points not greater than n (see [1, Theorems 4.1 and 4.2]).Comparing the two constructions, namely, of D(r , s) and of G n , it is evident that D(r , 0) is homeomorphic to G r for each integer r ≥ 3, (2.9) whence it follows from (2.7) that for every r ∈ {3, 4,...} and s ∈ {0, 1, 2,...,ℵ 0 } the dendrite G r is contained in D(r , s) even in such a way that E(G r ) ⊂ E(D(r , s)).
The next result follows from [1, Theorem 3.3] which gives a characterization of dendrites with a closed set of end points.But it is also a direct consequence of the definition of D(r , s). has a negative answer (this is the main result of [9]).Note that it is not the case for D(ℵ 0 , 0) because (by (3.1.1)above) the union U of Proposition 3.3 cannot be openly mapped onto D(ℵ 0 , 0).For further results in this direction see below.  p) for each p ∈ P and defined as a homeomorphism otherwise is obviously open.In particular, if P = R(D(ℵ 0 ,s)) and if t ∈ {1, 2,...,s} is the same fixed number for all ramification points p, then Y = D(ℵ 0 ,t), and the following proposition is obtained.Proposition 3.6.For each s ∈ {2, 3,...} and for each t ∈ {1, 2,...,s} there is an open mapping of D(ℵ 0 ,s) onto D(ℵ 0 ,t).
Taking t = 1 in the above construction we conclude from Proposition 3.2 that for each s ∈ {2, 3,...} there is no open mapping from D(ℵ 0 , 1) onto D(ℵ 0 ,s).Therefore the next result follows.We now consider open images of other members of Ᏺ, namely of dendrites D(r , s) for r ∈ {3, 4,...} and s ∈ {0, 1, 2,...,ℵ 0 }.To see that no one of them is minimal in the class Ᏸ ≤ O we need some facts about the structure of the set of end points of D(r , s).To this aim represent D(r , s) as in (2.2) and observe that if r = ℵ 0 , then the set we see that R(D(r , s)) = {R n : n ∈ N}, and that the sets R n are mutually disjoint.For each point q ∈ R n let F n s (q) denote, as previously, the union of s end free arcs in D(r , s) every two of which have the point q in common only.Since according to (2.5) the remainder W consists of end points of D(r , s) only, we have Observe that, simply by the construction, each point of E(F n s (q)) is an isolated point of E(D(r , s)).Further, since G r is homeomorphic to D(r , 0) ⊂ D(r , s) by (2.7) and (2.9) and since E(G r ) is homeomorphic to the Cantor set according to [1, Theorem 4.1], it follows again from (2.9) that W is homeomorphic to the Cantor set.
(3.3)Note further that if K is a component of the set cl(K) is an interior free arc of D(r , s) with one end point in R n and the other in R n+1 .(3.6) Therefore D(r , s) can be written as the following union: Proof.Fix r and s as assumed.Let D(r , s) be defined as the inverse limit by (2.2) and let, as previously, v be the only ramification point of the fan X 1 .For each n ∈ N choose a ramification point p n ∈ R n ⊂ D(r , s).Thus p 1 = v and p n ∈ vp n+1 for each n ∈ N. Then the sequence {p n } is convergent, and its limit e 0 = lim p n is, according to Statement 2.3, an end point of D(r , s) lying in the set W . Thus all points p n lie in the arc ve 0 , and if < is the natural ordering of ve 0 from v to e 0 , then • for each F n s (q) with q ∈ R n for some n ∈ N the restriction g | F n s (q) : F n s (q) → F n s (p n ) is a homeomorphism; • g(W ) = {e 0 }.By (3.7) the mapping g is well defined.It can be verified that g is the needed open retraction.

Propositions 3 .
2, 3.3, and 3.4 describe open images of D(ℵ 0 ,s) for s ∈ {0, 1, ℵ 0 }.For s ∈ {2, 3,...} some open images of D(ℵ 0 ,s) can be obtained in the following way.Fix any nonempty subset P ⊂ R(D(ℵ 0 ,s)).For any ramification point p ∈ P let F s (p) ⊂ D(ℵ 0 ,s) be the union of s end free arcs pe p 1 ,pes p 2 ,...,pe p s , every two of which have the singleton {p} in common only.Further, for a fixed t ∈ {1, 2,...,s} let F t (p) ⊂ F s (p) be the union of t end free arcs pe p i 1 ,pe p i 2 ,...,pe p i t .Then there is an open surjective mapping f (p) : F s (p) → F t (p) which is a homeomorphism on each free arc pe p j for each j ∈ {1, 2,...,s} with f (p) (p) = p and f (p) (e p j ) = e p i j for some i j ∈ {i 1 ,i 2 ,...,i t }.Then the mapping f : D

. 8 )
Further, for each n ∈ N take F n s (p n ) ⊂ X n ⊂ D(r , s) and note that ve0 ∩F n s (p n ) = {p n }.Put Y = ve 0 ∪ F n s p n : n ∈ N ⊂ D(r , s). (3.9) Define a mapping g : D(r , s) → Y such that • g | Y : Y → Y is the identity; • for each component K = ab of S as in (3.4), if a ∈ R n and b ∈ R n+1 , where n is determined by (3.5), the restriction g | K : K → p n p n+1 ⊂ ve 0 is a homeomorphism such that g(a) = p n and g(b) = p n+1 ;