IMFM
Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia
Preprint series Vol. 50 (2012), 1169
ISSN 2232-2094
WAZEWSKI'S UNIVERSAL DENDRITE AS AN INVERSE LIMIT WITH ONE SET-VALUED BONDING FUNCTION
Iztok Banic Matevž Crepnjak Matej Merhar Uros Milutinovic Tina Sovic
Ljubljana, January 27, 2012
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Wazewski's universal dendrite as an inverse limit with one set-valued bonding function
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Iztok Banič, Matevž Crepnjak, Matej Merhar, Uroš Milutinovic and Tina Sovič
January 20, 2012
Abstract
We construct a family of upper semi-continuous set-valued functions f : [0,1] ^ 2[0'1] (belonging to the class of so-called comb functions), such that for each of them the inverse limit of the inverse sequence of intervals [0,1] and f as the only bonding function is homeomorphic to Wazewski's universal dendrite. Among other results we also present a complete characterization of comb functions for which the inverse limits of the above type are dendrites.
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1 Introduction
In 1923 T. Wazewski described an example of a dendrite in the plane which contains a topological copy of any dendrite [26]. The described dendrite is now known as Wazewski's universal dendrite. In [20, p. 181] one can find a construction of Wazewski's universal dendrite using inverse limits. In particular, it is constructed as the inverse limit of an inverse sequence of planar dendrites and monotone bonding mappings. In this paper we con-
struct Wazewski's universal dendrite as the inverse limit of inverse sequence
of closed unit intervals [0,1] and a single upper semi-continuous set-valued bonding function. This new presentation of Wazewski's universal dendrite significantly simplifies the construction described in [20].
2010 Mathematics Subject Classification: Primary 54F50; Secondary 54C60. Key words and phrases: Continua, Inverse limits, Upper semi-continuous functions, Dendrites, Wazewski's universal dendrite.
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2 Definitions and Notation
Our definitions and notation mostly follow Nadler [20] and Ingram and Mahavier [13].
A map is a continuous function. For i = 1, 2, ni : [0,1] x [0,1] ^ [0,1] denotes the i-th projection from [0,1] x [0,1] onto the i-th factor, and for any positive integer i, pi : n^c=1[0,1] ^ [0,1] denotes the i-th projection from n~=1[0,1] onto the i-th factor.
A continuum is a nonempty, compact and connected metric space. A Peano continuum is a locally connected continuum.
A dendrite is a Peano continuum which contains no simple closed curve. Let D be a dendrite, b G D, and B a cardinal number. We say that b is of order less than or equal to B in D, written ord(b, D) < B, provided that for each open neighborhood U of b in D, there is an open neighborhood V of b in D, such that b G V C U and | Bd(V)| < B. We say that b is of order B, ord(b, D) = B, provided that ord(b, D) < B and ord(b, D) ^ a for any cardinal number a < B.
Points of order 1 in a dendrite D are called end points of D, the set of all end points of D is denoted by E(D). Points of order n > 2 are called ramification points and the set of all ramification points of D is denoted by R(D).
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A free arc in a dendrite D is an arc such that all its points but its end points are of order 2 in D. In particular, a maximal free arc in a dendrite D is an arc A with end points x and y in D such that Aif(E(D)UR(D)) = {x, y}. A continuum S is a star if there is a point c G S such that S can be
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presented as the countable union S =1) Bn of arcs Bn, each having c as
n=1
an end point and satisfying lim diam(Bn) = 0, such that Bn fi Bm = {c}
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when m = n. The point c is uniquely determined and is called the center of S. The arcs Bn are called beams of S.
Let D1 be a star in a compact metric space X. Let cA G R(D1) denote a point in the maximal free arc A, for each maximal free arc A of D1 (here maximal free arcs are precisely the beams of D1). Let C1 = {x1,x2,x3,...} be any subset of the set {cA | A is a maximal free arc in D1}. For each positive integer i, form
a star Si
disjoint from D1, making sure that Si If Sj = 0 only when i = j and that
lim diam(Si) = 0. Let D2 = D1 U (U°=1 Si). Next define D3 in a similar i—^
manner. Let ca G R(D2) denote a point in the maximal free arc A in D2,
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for each maximal free arc A of D2. Let C2 = {x1; x2, x3,...} be any subset of
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Theorem 2.1. For each positive integer n, Dn is a dendrite.
the set {ca | A is a maximal free arc in D2}. For each positive integer i, form a star Si
in X with the center xi and otherwise disjoint from D2, making sure that Si H Sj = 0 only when i = j and that lim diam(Si) = 0. Let
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D3 = D2 U (USi). Continuing in this manner, we obtain a continuum Dn for each positive integer n. The following theorem (already implicitly used in the above inductive construction) is a well-known fact, see [20] for details.
The construction of the continuum, homeomorphic to Wazewski's universal dendrite in [20, p. 181] uses the abovementioned construction of a
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chain of dendrites D1 C D2 C D3 C ..., then defines certain bonding maps fn : Dn+1 ^ Dn, and then finally obtains Wazewski's universal dendrite as Jm{Dfc ,ffc }«=i.
Finally we state a result that is characterizing Wazewski's universal dendrite that will be needed in Section 4.
Theorem 2.2. For any dendrite D, D is homeomorphic to Wazewski's universal dendrite if and only if its set of ramification points is dense in D and each of its ramification points is of infinite order.
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Proof. [7, p. 169], [26, p. 123]	□
If (X, d) is a compact metric space, then 2X denotes the set of all nonempty closed subsets of X. Let for each e > 0 and each A G 2X
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Nd(e, A) = {x G X | d(x, a) < e for some a G A}.
The set 2X will be always equipped with the Hausdorff metric Hd, which is defined by
K	) = inf{e > 0 | H C Nd(e,K),K C Nd(e, H)},
for H, K G 2X. Then (2X, Hd) is a metric space, called the hyperspace of the space (X, d). For more details see [12, 20].
When we say that f is a set-valued function from X to Y, we mean that f is a single-valued function from X to 2Y, i.e. f : X ^ 2Y. By a slight misuse of notation and terminology we will also say that function f : X ^ 2y is set-valued (without explicitly mentioning "from X to Y"). A function f : X ^ 2Y is surjective set-valued function from X to Y if
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for each y G Y there is an x G X, such that y G f (x).
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The graph r(f) of a set-valued function f : X ^ 2Y is the set of all points (x,y) £ X x Y such that y £ f (x).
A function f : X ^ 2Y, where X and Y are compact metric spaces, is IN	upper semi-continuous set-valued function from X to Y (abbreviated u.s.c.)
if for each open set V C Y the set {x £ X | f (x) C V} is an open set in X.
The following is a well-known characterization of u.s.c. functions between metric compacta (see [13, p. 120, Theorem 2.1]).
Theorem 2.3. Let X and Y be compact metric spaces and f : X ^ 2Y a set-valued function. Then f is u.s.c. if and only if its graph r(f) is closed in X x Y.
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An inverse sequence of compact metric spaces Xk with u.s.c. bonding functions fk is a sequence {Xk, fk}£=1, where fk : Xk+1 ^ 2Xk for each k.
The inverse limit of an inverse sequence {Xk, fk}£=1 with u.s.c. bonding functions is defined to be the subspace of the product space Xk of all x = (x1, x2, x3,...) £	Xk, such that xk £ fk(xk+1) for each k. The
inverse limit of {Xk, fkis denoted by ^im{Xk, fk}£=1.
Note that each inverse sequence {Xk, fk}£=1 with continuous single-valued bonding functions can be interpreted as an inverse sequence with 2 u.s.c. set-valued bonding functions and that the inverse limits obtained according to both interpretations coincide. Therefore we do not specially emphasize the status of bonding functions in inverse sequences we deal with.
The notion of the inverse limit of an inverse sequence with u.s.c. bonding functions was introduced by Mahavier in [18] and Ingram and Mahavier in [13]. Since the introduction of such inverse limits, there has been much interest in the subject and many papers appeared [1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 16, 21, 22, 23, 24, 25].
The most important case in the present paper is the case when for each k, Xk = [0,1] and fk = f for some f : [0,1] ^ 2[0'1]. In such case the inverse limit will be denoted by ^im{[0,1], f }£=1.
On the product space Xn, where (Xn,dn) is a compact metric space
n=1
for each n, and the set of all diameters of (Xn, dn) is majorized by 1, we use the metric
s	J dra(xra,yra)
D(x,y) = sup
n€{1,2,3,...} I 2n
where x = (x1, x2,x3,...), y = (y1,y2 ,y3,...). It is well known that the
metric D induces the product topology [10, p. 190].
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3 The comb functions
Let A C [0,1] x [0,1] be defined by
A = {(t,t) G [0,1] x [0,1] | t G [0,1]}.
For any positive integer n, let {(a», bi)}™=i be a finite sequence in [0,1] x [0,1], such that a» < 6» for each i = 1, 2, 3,..., n and a» = aj whenever i = j. Next denote by A(aj,6j)™=1 the union
n
A(aj, bj)™=i = U^A] x {a»}) C [0,1] x [0,1].
j=i
Then
G(aj, bj)n=i = A U A(aj,bj)n=i
is closed in [0,1] x [0,1], since it is a union of finitely many closed arcs. Furthermore ni(G(aj, bj)n=i) = n2(G(aj, bj)n=i) = [0,1]. Therefore by Theorem 2.3 there is a surjective u.s.c. function /(ai,bi)n_1 : [0,1] ^ 2[0'i] such that its graph r(/(„.A)n=l) equals to G(aj,6j)n=i.
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Definition 3.1. Let n be a positive integer and {(a», bj)}j=i be a subset of [0,1] x [0,1], such that 0 < a» < 6 for each i = 1, 2, 3,..., n and a» = aj whenever i = j. Then / : [0,1] ^ 2[0'i] is called an n-comb function with
resPect to	if / = /(ai,biYU.
We also say that / : [0,1] ^ 2[0'i] is an n-comb function, if / is an n-comb function with respect to some {(a», 6»)}™=^
Figure 1: The graph of an 8-comb function
It is not necessary to eliminate the possibility a» = 0 for some i (all the proofs in the paper would go through also in such case), but we have chosen
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to do so in order to reduce the number of cases that must be examined in the proofs and since the main result can be obtained with this restriction in place.
Definition 3.2. Let for each j, j be a nonnegative integer. We use
(a? ,42,43,...)
to denote the point (ai, a?,..., a?, a2, a2, • • •, a2,... ) and
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(ai1 ,a22 ,a33,..., aj ,t~) to denote the point (a?, a?,..., a?, a2, a2,..., a2,..., aj, aj,..., a j, t, t, t, • • •).
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Example 3.3. Let f be a 1-comb function with respect to {(ai, bi)}i=1. Then x G ^im{[0,1], f }£=? if and only if
1.	either x = (t~) for a t G [0,1] or
2.	there is a positive integer n such that x = (an, t^) for a t G (a?, b?].
Therefore l^m{[0,1],f }£= ? is the star with the center (a^°) and beams Bo = {(t~) | t G [0, a?]}, B0 = {(t~) | t G [a?, 1]} and Bn = {(a?,t~) | t G [a?, b?]}, n = 1, 2, 3,...
Figure 2: The graph of an 1-comb function and its inverse limit
Example 3.4. Let f be a 2-comb function with respect to {(ai, b^}2^,
where a? < a2. We distinguish the following two cases: 1. b? < a2
Then x G ^im{[0,1], f }£=? if and only if (a) either x = (t~) for a t G [0,1] or
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(b) there is a positive integer n such that x = (an,tf) for a t G (a1, b1]
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or
(c) there is a positive integer n such that x = (an,tf) for a t G (a2,b2].
Therefore l^m{[0,1],f}f= 1 is the union of two stars. The star S with the center (af) and beams Bo = {(tf) | t G [0, a1 ]}, B0 = {(tf) | t G [a1,1]} and Bn = {K,tf) | t G [a1,b1]}, n = 1, 2, 3,..., and the star S0 with the center (af) and beams Cn = {(an,tf) | t G [a2,b2]},
n
1, 2, 3,
Figure 3: The graph of a 2-comb function and its inverse limit, b1 < a2 2. b1 > a2
Then x G ^m{[0,1], f }f=1 if and only if
(a)	either x = (tf) for a t G [0,1] or
(b)	there is a positive integer n such that x = (an,tf) for a t G (a1, b1] or
(c)	there is a positive integer n such that x = (an,tf) for a t G (a2, b2] or
(d)	there are positive integers n and m such that x = (an,am,tf) for a t G (a2, b2].
Therefore l^m{[0,1],f}f 1 is the union of countable many stars. The star S with the center (af) and beams B0 = {(tf) | t G [0,a1]}, B0 = {(tf) | t G [a1,1]} and Bn = {K,tf) | t G KA]}, n = 1, 2, 3,..., the star S0 with the center (af) and beams Cn = {(an,tf) | t G [a2,b2]}, n = 1, 2, 3,..., and for each positive integer k the star Sk with the center (a^, af) and beams C^ = {(a1 ,an,tf) | t G [a2,b2]},
n
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Figure 4: The graph of a 2-comb function and its inverse limit, b1 > a2
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Figure 5: The graph of a 2-comb function and its inverse limit, bi = a2
Note that if b1 = a2 the stars Sk, k = 1, 2, 3,..., are attached at the end points (a^, ) of S and if b1 > a2 the stars Sk, k = 1, 2, 3,..., are attached at the interior points of the maximal free arcs {(a^, t^) | t G [a1,b1]} of S, k = 1, 2, 3,...
In the following theorem we show that any inverse limit of intervals [0,1] and a single n-comb function is a dendrite.
Theorem 3.5. Let n be any positive integer and let f : [0,1] ^ 2[0'1] be any n-comb function. Then hm{[0,1], f }£=1 is a dendrite.
Proof. We prove Theorem 3.5 by induction on n by proving the more precise claim that includes also information about maximal free arcs and ramification points in the dendrite. For each positive integer I, let us introduce the following notation for certain statements that will be used in the inductive proof of the theorem:
(a)^ The inverse limit D£ = hm{[0,1], f(ai,&i)i_1 }fc=1 is a dendrite.
(b)^ The points of the form (x1,x2,x3,... ,a°°) G D^, j < I, are exactly the ramification points of D^.
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(e)^ The maximal free arc in Dl having the point x = (xi, x2, x3,..., xm, b°°) described in (c)^ as one endpoint, has (x1, x2, x3,..., xm, ) as the
(c)^	The points of the form (x1, x2, x3,..., xm,b°°) G D^ i < I, where m > 1, aj = xm = bj, and b G {ai+1, aj+2, aj+3,..., a^j, are endpoints of Di.
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(d)l	All endpoints of Dl are of such form, except endpoints (0°°) and (1°°).
described in (c)i as one endpoint, has (X'1, x2, x3,. . . , xm,«i £ other endpoint if ai < fc; if ai > bj then the maximal free arc in Di
ending at x equals to the maximal free arc in Dl-1 ending at x. (f)i The arc with endpoints (a°°) and (1°°) is a maximal free arc in Dl.
1. Let n =1. There are a1,b1 G [0,1] such that a1 < b1 and / = /(«,bi)i=1. In Example 3.3 it was shown that the inverse limit D1 = hm{[0,1],/j°=1 is a star, and is therefore a dendrite. We see that (a °°) is the only ramification point of D1 and that maximal free arcs of D1 are exactly the beams Bo = {(t°) | t G [0, a1] j, B0 = {(t°) | t G [a1,1]j and Bk = {(ak,t °) | t G [a1,b1]j, k = 1, 2, 3,... of the star
2. Let / be any n-comb function, n > 2. Without loss of generality we
D1. Note that (a)1-(f)1 hold true.
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may assume that / = /(ai,bi)n_1, where a1 < a2 < a3 < ... < an. Let, as the inductive assumption, (a)n-1-(f)n-1 hold true for /(«)bi)n-i. We show that the inverse limit
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lim{[0,1], /}°=1 = Dra = lim{[0,1], /k^ jk=1
satisfies all the abovementioned properties for I = n.
By the inductive assumption Dn-1 = ^im{[0,1], /(« ^n-i j °=1 is a dendrite.
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Case 1. an > bj for each i = 1, 2, 3,..., n — 1
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In this case any x G Dn \ Dn-1 is of the form x = (a^j, t°°), where k is a positive integer and t G (an , bn]. Therefore
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Dn = Dj-1 U S,
where S = {(an,t°°) | k G N, t G [an,bn]j, and we see that S is a star with the center (a°°) G Dn \ R(Dn-1). Obviously (a)n-(f)n hold true.
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Case 2. an < bj for some i = 1, 2, 3,..., n — 1
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In this case we show that
Dn = Dra-1 U (|JS
where
(a) S = {S1; S2, S3,...} is a countable family of stars with centers c1,c2, c3, ... respectively, where c1,c2,c3, ... G Dn\R(Dn-1), and each of the maximal free arcs in Dn-1 contains at most one of these centers,
(b)	for each positive integer i, Si H Dn-1 = {c },
(c)	Si H Sj = 0 if i = j, and
(d)	lim diam(Si) = 0,
and therefore it will follow that Dn is a dendrite by Theorem 2.1, using (a)n-1. That will prove (a)n.
Any point of Dn\Dn-1 is of the form (x1; x2, x3,... , xm, an, t^), where k is a positive integer, m is a nonnegative integer, t G (an,bn], and xm = an, and vice versa.
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The set
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{(X1,X2,X3, . . . ,xm,an,t~) | k > 1, t G [an,bn]} is a star centered in (x1; x2, x3,..., xm, a^°) having the beams {(X1,X2,X3, . . . ,xm,an,t°°) | t G [an,bn]},
for each k > 1. Note that S is infinite since for each i such that an < bi the family S contains stars centered at (ak, ) for each positive integer k.
From f-1 ! (an) = {an} it follows that if for x G Dn-1 and for some positive integer m, pm(x) = an, then pm+1(x) = an. Therefore such x ends with the block . Let X1 = {(a^°)} and let for each positive
integer m > 2,
Xm = {x e Dn-1 | Pm(x) = ßn,Pm-l(x) = ßn}-
Then Xm is a finite set for each m. Therefore X = (Jm=1 Xm is a finite or countable infinite subset of Dn-1 \ R(Dn-1) ((b)n-1 is also used). Also, each maximal free arc of Dn-1 contains at most one x e X.
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To prove this, we shall for each x £ X find the uniquely determined maximal free arc of Dn-i containing x. Let
x = (x1, x2, x3,..., xm, a0) £ X,
where xm = an. Then xm = aj for some i < n. Note that since a £ f(ai bi)"-L1 (an), it follows that an £ [ai,bi] and therefore b > an. Now we distinguish two cases, b > an and b = an.
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If bj > an, then bj £ {aj+i, aj+2,..., a„}, hence (xi
, x2, . . . , xm* bj ) is
an endpoint of Dn_i by (c)n_i and the arc
{(xi,x2,x3,... ,xm,t00) | t £ [a„_i, bj]}
is a maximal free arc of Dn-i by (e)n-i. Obviously x belongs to the arc, since an £ [an-i,bj].
If bj = an, then x is an endpoint of Dn-i by (c)n-i, and clearly it belongs to the maximal free arc {(xi,x2,x3,...,xm,t00) | t £ [an-i,bj]} of Dn-i, which is a maximal free arc in Dn-i by (e)n-i.
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Now, when we have the explicit description of all maximal free arcs in Dn-i containing elements of X, we see that each such maximal free arc contains exactly one point from X.
Take any x = (xi,x2, x3,..., xm, a0) £ X, where xm = an. Then xm = aj for some i < n. For each positive integer k, let
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Bfc = {(xi,x2,x3,... ,xm,an,t00) | t £ [a„,b„]}.
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Obviously, is an arc in Dn and S(x) = (J0=i is a star centered at x. The diameter of S(x) satisfies
diam(S(x)) < D((xi, x2,...,xm, 000), (xi, x2,..., xm, 100)) <
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Since for each m there are only finitely many such points x £ X (Xm is finite), it follows that the set S = {S(x) | x £ X} is finite or it can be presented as S = {Si,S2,S3,...}. From the above upper bound for the diameters of the stars in the infinite case it follows that lim diam(Sj) = 0.
Take any point x £ Dn \ Dn-i. As already noticed, it is of the form
x = (xi,x2, x3,..., xm, aj;, t°°), where k is a positive integer, m is a
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nonnegative integer, t G (an,bn], and xm = an. Therefore x G S(y), where y = (xi	) G X. Therefore
Dn \ Dn-1 = ( U S(X)J \ X = (U s) \ X,
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and finally
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Dn = Dn-i U ( ^ S(X)) = Dn-i U (|J S) ,
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proving (a)n.
To prove that the points of the form x = (xi, x2, x3,... , xm, 6°°) G Dn, where i < n, xm = a», and 6» G {a»+i, a»+2, aj+3,..., an}, are endpoints of Dn, we distinguish two cases. If i < n — 1 then x G Dn-i and then x is an endpoint of Dn-i by (c)n-i, since 6» G {a»+i, a»+2, a»+3,..., an-i}. Since b = an the only star attached to the maximal free arc in Dn-i ending at x is centered at a point that differs from x, or no star is attached to that arc at all, it follows that x G E(Dn). If i = n then b = bn and therefore x is an endpoint of a star from S. That proves
(c)n.
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Also each endpoint of Dn, which belongs to Dn-i, is also an endpoint
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in Dn-i, therefore it is of the form x = (xi, x2, x3,..., xm, b°°) G Dn, where xm = b and b G {a^+i, a^, a»+3,..., an-i}, by (d)n-i. Points of such form with b» = an are centers of the newly attached stars and therefore are not endpoints of Dn. It follows that b = an and therefore b G {ai+1, a»+2, a»+3,..., an}. Each endpoint of Dn, which belongs to Dn \ Dn-i, is necessarily an endpoint of a newly attached star and therefore is of the form x = (xi, x2, x3,..., xm, b°°), an = xm = bn. Additional condition b G {ai+1, a»+2, a»+3,..., an} is satisfied vacuously for i = n. Obviously (0°) and (1°) are endpoints of Dn, too. That proves (d)n.
Let x = (x1, x2, x3,..., xm, b°°) G Dn be any endpoint of Dn mentioned in (c)n, where a» = xm = b and b G {«¿+i, «¿+2, «¿+3, • • •, «nj.
If i < n then by (c)n-i x is an endpoint of Dn-i. If an < b then we have already proved that a new star centered at (xi, x2, x3,..., xm, a°°) is attached to the maximal free arc of Dn-i ending at x, and since no other star was attached to this arc it follows that (xi, x2, x3,..., xm, a°°) is the other endpoint of the maximal free arc of Dn ending at x. If an > b then no star was attached to the maximal free arc of Dn-i
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ending at x = (xi , X2, X3,..., xm. and therefore it remained a maximal free arc of Dn as well. This proves (e)n.
By (f)n-1 the maximal free arc of Dn-1 having (1^) as one endpoint has (a^—1) as the other endpoint. Since a star centered at (a^°) was attached to Dn-1, and since no other star was attached to the above-mentioned arc, (f)n follows.
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Finally (b)n follows from (b)n-1 and from the fact that at each point of the form (x1, x2, x3,..., xm, G Dn a new star was attached to -1, as shown above.
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In the following remark we extract certain parts of the above proof for later use.
Remark 3.6. Let n be a positive integer.
1.	For each positive integer n and for each y G Dn, y is either of the form y = (t~), t G [0,1], or of the form y = (aj1, ag, , • • • , ofr, O, where m is a positive integer and for each I < m it holds that ^ < n, ki > 0,	< , and aim < t < bim.
2.	Any point of Dn+1 \ Dn is of the form
(x1j x2 j x3 j • • • j xmj «n+l' ^^
£
where k is a positive integer, m is a nonnegative integer, t G (an+1, bn+1], and xm = an+i.
HH
3.	x G Dn is a ramification point in Dn if and only if there are positive integers m and j < n, such that (x) = « for each positive integer k > m.
Definition 3.7. We will use Dn to denote the dendrite
Dn = Üm{[0, 1],/(ai;bi)n=i
Next we define functions that we shall use later in proof of the main result.
Definition 3.8. We define the function fn : Dn+1 ^ Dn by
&
t f \ gn(x) ; x G Cl(Dn+1 \ Dn),
fn(x) =
cu
x ; x G Dn,
o
CM
IN
CM ft
vD
£ CO CO
where : Cl(Dn+1 \ ^ is defined as follows. Any point of Cl(Dn+1 \ is of the form
x (x1 j x2 j x3 j * * * j xm j an+1' t ) '
where k is a positive integer, m is a nonnegative integer, t G [an+1,6n+1], and xm = an+1 (see Remark 3.6), and we define
gn(x) = (X1)X2)X3) * * * jXm,0^+1)*
fi
Note that is continuous for each n by [19, Theorem 7.3 p. 108].
Lemma 3.9. Let x G
1. If
X =(ak;jak^3,***,«j,0 G D
where j > 0, i1, i2, i3,..., ij < n, ail < ai2 < ■ ■ ■ < a^, k1, k2,..., kj > 0, and t £ [aj, bj ], then for each
y £ f—1 (x)
and for each i < k1 + k2 + k3 + ... + kj + 1 it holds that pi (x) = pi(y).
CM
2. If x = (t~), t £ [0,1], then for each
00 1
y £ fn (x)
CM
it holds that p1(x) = p1 (y) = t.
Proof. If y £ Dn, then y = x and the claim is obviously true. Note that in Case (1) from t = a^ it follows that y £ Dn.
If y £ Dn+1 \ Dn, then by Remark 3.6 (2) y is of the form
(x1
where k is a positive integer, m is a nonnegative integer, s £ (an+1,bn+1], and xm = an+1. Then x = f(y) = gn(y) = (x1,x2,x3,... ,xm,a£+1).
In Case (1) in the remaining subcase t = a^ it follows that m = k1 + k2 + k3 + ... + kj and t = an+1. Therefore (x1, x2, x3,..., xm, an+1) = CO (ak1, ak2, akj3,..., a§ ,t).
In Case (2) it follows that m = 0 and t = an+1.	□
G
Lemma 3.10. Let x = (x1, x2, x3,..., xm, a^°h1) £ Dn, where n is a positive integer, m is a nonnegative integer, and xm = an+1. Then f—1(x) is a star centered in x.
£ CO CO
CD u
and for each k the set
Proof. From what we have seen in the proof of Lemma 3.9 it follows that
o
fn^x) = (J {(x?,x2,x3, • • • ,xm,an+!, s°°) | s G [an+?, bn+^}, k=?
Bk = {(x?,x2,x3, • • • , xTO, an+?, s°°) | S G [an+?,bn+?]} is an arc with endpoints x and (x?, x2, x3,..., xm, an+?, bn+?), Bi HBj = {x}
Ctf
Let {(an,bn)}n=? be any sequence in [0,1] x [0,1], such that an < b for each positive integer n, and ai = aj whenever i = j. Next denote by A(an,bn)n=? the union
for any i = j, and lim^^, diam(Bk) = 0.	□
oo
n bn)}n=? be any sequence in [o, 1] x [o, l], such that an < bn
oo
A(an,bn= U(K,bn] x {an}) C [0,1] x [0, 1].
n=!
and by G(an, bn)n= the subset of [0,1] x [0,1], defined by
G^nA^ = A U A^nA)^,
where A = {(t,t) | t G [0,1]} as above.
It is easy to see that n?(G(ai, bi)n=?) = n2(G(ai, bi)n=?) = [0,1].
Obviously G(an,bn)n= is not necessarily closed in [0,1] x [0,1]. The following theorem gives a whole family of sets G(an, b^m that are closed in [0,1] x [0,1].
Theorem 3.11. Let {(an,bn)}n= be any sequence in [0,1] x [0,1], such that
1.	an < bn for each positive integer n,
2.	ai = aj whenever i = j,
3.	lim (bn — an) = 0.
n^oo
I
Then G(an, bn)n= is a closed subset of [0,1] x [0,1].
CO
Proof. Let {xn}n= be any sequence in G(an, bn)n=?, which is convergent in [0,1] x [0,1] with the limit x0 G [0,1] x [0,1]. We show that x0 G G(an, b^^.
If there are positive integers k and n0, such that xn G [ak,bk] x {ak} for each n > n0, then, since [ak, bk] x {ak} is compact, x0 G [ak, bk] x {ak}
and therefore x0 G G(an,bn)n=?. Otherwise there are strictly increasing

o
£ CO CO
sequences {¿n}nL1 and {jn}n=1 of integers, such that xin G ([ajn, bjn] x
l"nJn=1	\JnJn=1
{ain}) U A, where A = {(t,t) G [0,1] x [0,1] | t G [0,1]}, for each positive integer n. Since lim (bn — an) = 0, it follows that x0 G A and therefore
n—
xo G G(an,bn)n=1.	□
Therefore by Theorem 2.3 it follows that for any sequence {(an,bn)}n=1 satisfying
1.	an < bn for each positive integer n,
2.	ai = aj whenever i = j,
3. lim (bn — an) = 0,
n—x
there is a surjective u.s.c. function f(a„,&n)n_1 : [0,1] ^ 2[0'1] such that its
(an,bn)cr
graph r(f(„n;bn)n_1) equals to G(an,bn)n=1, since G(an,bn)n=1 is a closed
subset of [0,1] x [0,1] by Theorem 3.11, and since
n1(G(ai,bi)n=1) = n2(G(ai,bi)n=1) = [0,1].
Definition 3.12. Let {(an,bn)}n=1 be any sequence in [0,1] x [0,1], such that
i
CNI
1.	an < bn for each positive integer n,
2.	ai = aj whenever i = j,
3. lim (bn — an) = 0.
n—x
Then f(an,bn)n_1 is called the comb function with respect to {(an, bn)}n=1.
We also say that f : [0,1] ^ 2[0'1] is a comb function, if f is the comb function with respect to some sequence {(an,bn)}X=1 in [0,1] x [0,1] satisfying (1), (2), and (3).
Theorem 3.13. Let f : [0,1] ^ 2[0'1] be the comb function with respect to
the sequence {(an,bn)}n=1. Then
CD ■ i Jh CD CO
hm{[0,1],f}X=1 = ci U dJ .
Vn=1
Proof. Obviously, since hi—{[0,1], f }X=1 is closed in n=1[0,1]
Jh
/ oo
lim{[0, 1], f}?°
cu
him{[0,1],f }sr=1 d ci |J Dn
n=1
CO
Let x e Tjm{ [0,1], f }°°=1 \ U^=1 Dn. Obviously x is of the form
X	(ßii , ai2 , ai3 ; - - -) ,
Next we show that
o
Hm{[0,1],f}r=1 C Cl |J D
\n=1
where {ain | n =1, 2, 3,...} is an infinite subset of {an | n = 1, 2, 3,...}.
Take any open ball U = B(x,e) in n=1[0,1] with respect to the metric D. Let m be a positive integer such that ^m < e. Then
vO
(aii ,ai2 ,...,aim_i ,aim ) G U H Dim .
1—1
C	CO
In the above proof we noticed that any x G lim{[0,1], f }k=1 \ U^=1 Dn is
of the form x = (ai1, ai2, ai3,...), where {ain | n =1, 2, 3,...} is an infinite
subset of {an | n = 1, 2, 3,...}. We can make this statement more precise
taking into account the definitions of inverse limits and comb functions as
follows:
00
Remark 3.14. Let f : [0,1] ^ 2[0'1] be the comb function with respect to the sequence {(an, bn)}n=1. Any point x G Hm{[0,1], f }k=1 \ Un=1 Dn is of
"nJJn=1- 1 ^J	^ ^ l— IL^-M) J Jfc=1 \ Un=
the form
K1 ,«22 ,«33 ,-••),
~i1 '	' i3
where for each I it holds that k > 0 and ai£ < ai£+1 < bi£.
In Examples 3.15 and 3.16 we show that there are comb functions f, such that the inverse limits lim{[0,1], f }k=1 are not dendrites.
Example 3.15. Let (a1; b1) = (1, 1) and let for each positive integer n > 2, (ßn,bn) = (2 - n+1 > 2 + n+1). We show that l—{[0,1]>fK,M~_i}°=is not
locally connected and therefore it is not a dendrite. Let
CD
1 1
xo = k,1, 1°°) e lim{[0,1], fK^)So_i}°== 2 2
and e = min{d(xo,K), ^} > 0, where K = {(t°) | t e [0,1]}. Let r < e and y = (y1,y2,y3, • - •) e B(xo,r) H lm{[0,1], f(an,fcn)~_1 }°=1 be arbitrarily chosen. Then, since r > D(x0,y) > -1—3, it follows that y3 > 1 — 23r.
o
IN
b &
VD
0
o
1
CO
£ CO CO
Figure 6: The graph of the comb function from Example 3.15
U
K
Figure 7: The continuum from Example 3.15
CO CD
$H
CD CO
u
a
CD U
cu
Therefore y3 > 1 — 23r > 1 — = | and hence y^ Furthermore G f (y?) = {1 ,y3}. If y2 = y3 then
y3 for each i > 3.
y2
>
1
= 12 >r'
22 22 2 and hence yi G f ( 1 ). Clearly there is a
D(xo,y) >
a contradiction. Therefore y2 = positive integer n, such that yi ^^ 2	2
Therefore for each r < e, y G B(x0,r) if and only if there is a positive integer n, such that y = (an, 1, t00), where 2 "" < r and t > 1 — 23r.
1 2 — an and —
2
xi-yi
< r.
Therefore for each r < e the intersection B(x0, r)fl^im{[0,1], f(„n}
oo fc=1
CM
o
CM
IN
CM ft
a
et vD
0
£ ^
&
O
CM
1
CM 00 CM CM
£ CO CO
CO CD
$H
CD CO
$H
a
CD Jh
cu
is the union of countably many mutually disjoint intervals
1
{(«„,-, t°°) | t G (1 - 23r, 1]},
where
< r. See Fig. 7.
Example 3.16. Let (a1, b1) = (2, 1) and let for each positive integer n > 2, (an, bn) = (2 — n, 2). A similar argument as in Example 3.15 shows that the
Figure 8: The graph of the comb function in Example 3.16
inverse limit ^im{[0,1], /(an,bn)^_1 }fc=i is not locally connected and therefore it is not a dendrite.
In Theorem 3.20 we prove that under rather weak additional assumptions the inverse limit of a comb function is a dendrite. Essentially, the conditions say that the only comb functions for which the inverse limits are not dendrites are similar to those from Examples 3.15 and 3.16. Before stating and proving the theorem we introduce the necessary notion of admissible sequences and prove a few lemmas.
Definition 3.17. The sequence {(an,bn)}n=1 in [0,1] x [0,1] is admissible if for each positive integer n there is a positive integer ^(n) > n, such that for each m > ^(n) it holds that if am < an, then < an.
Lemma 3.18. Let / : [0,1] ^ 2[0'1] be any comb function with respect to a sequence {(an,bn)}n=1, and let
x (aki , at2 , ,
•,afj,t°°) G Dn
1 -an
2
2
CM
i-H
o
CM

j > 0, ii,i2,i3,... ,ij < n, a»1 < ai2 < ••• < a^, fci,fc2,...,fcj- > 0, and t G [a»., b».]. Let / be the functions defined in Definition 3.8. Then for each
y G Cl(U(/n
CM
fc>n
and for each i < ki + k2 + k3 + ... + kj + 1 it holds that p»(x) = p»(y) (where x and y are interpreted as elements of n°=i[0,1]).
Proof. By induction on k — n we prove the following claim: For each
y G (/n o /n+i o ... o ffc)-i(x)
and for each i < ki + k2 + k3 + ... + kj + 1 it holds that p»(x) = p»(y).
For k — n = 0 the statement holds true by Lemma 3.9 (part (1) for j > 0 and part (2) for j = 0).
Let k — n = I and assume that the claim is true for l — 1. Since
(/n ◦ /n+i ◦ ... ◦ / )-i(x)=	|J	/fc-i(z)
ze(/nO/n + lO.,.o/fc_i)-1(x)
for any y G (/n o /n+i o... o /) i(x) we choose z G (/n o /n+i o... o	i(x)
such that y G /fc_i(z). By the induction assumption p»(x) = p»(z) for each
i < ki + k2 + k3 + ... + kj + 1 and by Lemma 3.9 p»(y) = p»(z) again for each i < ki + k2 + k3 + ... + kj + 1. This completes the proof since the limits of sequences of points with the required property have the property.	□
£
We will also need the following lemma about point preimages.
CO
Lemma 3.19. Let / : [0,1] ^ 2[0'i] be the comb function with respect to any admissible sequence {(an, bn)}°=i. For each e > 0 there is a positive integer k such that
diam( U (/ ◦ /fe+i ◦ ... ◦ /n)-i(p)) < e
n>fc
m
for each p G Dk, where maps /n are defined as in Definition 3.8.
•
Jh
Proof. Let e > 0 and m be a positive integer such that < e. Also let n0 > m be any positive integer such that for each n > n0, it holds that bn — an < mm. For each positive integer l, let ^(l) be a positive integer such that for each n > ^(l) it holds that if an < a^, then bn < a^ (here we use the admissibility of the sequence {(an,bn)}°°=i). Let
CV
i-H
o
IN
CO
k0 = max{n0,^(1), ^(2),^(3),...,^(n0)}, k? = max{n0,^(1), ^(2),^(3),...,^(k0)}, k2 = max{n0,^(1), ^(2),^(3),..., ^(k?)},
km = max{n0,M1),M2),M3),... ,Mkm_1)}.
Then we show that
a
cd
is a positive integer, such that
k = max{n0,^(1),^(2),^(3),... ,^(km)}
diam( U (fk o fk+! o ... o fn) ?(p)) < £ n>k
1—1
for each p G Dk.
Take any p G Dk. Then by Remark 3.6 (1) p is either of the form p = (t 00), t G [0,1], or of the form p = (p^p2,p3,... ,pj,t°°), where pj = as for some s < k and t G (as, bs]. Clearly, it holds that
o
diam(U (fk o fk+? o ... o fn)_?(p)) < diam(U (fk o fk+1 o ... o fn)_1(t00))
n
n> k	n> k
since
(fk o fk+1 o ... o fn)_? ((p1, p2, p3, . . . ,pj, t °°)) = ^ _
{(p!,p2,p3, . . . ,pj ,x?,x2,x3, . . .) | (x?,x2,x3, . . .) G (fk ofk+?o. . .ofn) 1 (t 0)}
If t = ai for all i > k, then Un>k(fk o fk+1 o ... fn)_1(t00) = {(t°)} and therefore diam(Un>k(fk o fk+1 o ..". fn)_1(t°)) = 0. If t = ai for some i > k, then we shall prove that
diam^ (fk o fk+1 o ... o fn)_1 (t 00)) < £
n>k
CO
by proving that
£
D(y, (t°)) = D(y, (af)) < 2 for arbitrary y = ^1^2^ . . .	. . .) G Un>k(fk o fk+1 o ... o fn)_1(t0).
Since y? = t = ai by Lemma 3.18 (and therefore yi_ai = 0), and since y? < y2 < y3 < ..., it follows that
a
,	ry2 — ai y3 — ai ym — ai 1 ,
D(y, (a0°)) < supr o i, Q i,..., —-i, —¡r}.
\ni \ i )) — f i 22 23	2m 2m+1
CM
Let j £ {2, 3,4,..., m} be arbitrary. We show that
o
yj - a < £
~ 2j 2.
IN
CM £
First we show that for each s £ {2, 3, 4,..., j} there is a positive integer I > n0 such that ys,ys-1 £ [al,bl].
For s = 2, the claim is true since y2 £ [ai,bi], y1 = a», and i > k > n0. If y2 / {an | n = 1, 2, 3,...}, then y2 = y3 = y4 = ■ ■ ■ and therefore for each s £ {3,4,5,..., j}, ys = = y2 £ [a*,b*].
In the rest of the proof we consider the case y2 = ai0 for some positive integer i0. If i0 < km, then ^(i0) £ {n0,^(1), ^(2),^(3),..., ^(km)} and therefore ^(i0) < k. Since k < i, it follows that ^(i0) < i. Therefore from y2 = ai0 > a* it follows that ai0 > b* and this contradicts the fact that ai0 = y2 £ [ai,bi]. So in this case i0 > km > n0 and the claim for s = 3 follows, since y3,y2 £ [a*0, b*0].
If y3 / {an | n = 1, 2, 3,...}, then y3 = y4 = y5 = ■ ■ ■ and therefore for each s £ {4, 5, 6,..., j}, ys = ys_1 = y3 £ [a^, b^].
In the rest of the proof we consider the case y3 = ail for some positive integer i1. If i1 < km-1, then ^(i1) £ {n0, ^(1), ^(2), ^(3),..., ^(km-1)} and

vD
O
O
therefore ^(i1) < km. Since km < i0, it follows that ^(i1) < i0. Therefore
i
from y3 = ail > ai0 it follows that ai1 > bi0 and this contradicts the fact that ail = y3 £ [ai0, bi0]. So in this case i1 > km-1 > n0 and the claim follows for s = 4 since y4,y3 £ [ail,bil].
Repeating this reasoning m times proves that for each s £ {2, 3, 4,..., j} there is a positive integer I > n0 such that ys,ys-1 £ [a^b^]. It follows that
CM CM
£ CO CO
d(yj,a*) < d(yj,yj-1) +... + d(y3,y2) + d(y2,a*) < (j - < £
m
since ys,ys-1 £ [a^b^] for each s, for some I > n0. Therefore jm1 < f. D
Theorem 3.20. Let f : [0,1] ^ 2[0'1] be the comb function with respect to any admissible sequence {(an, bn)}n=1. Then ]^im{[0,1], f }£=1 is a dendrite.
CD
Proof. We show that l]m{[0,1], f }£= 1 is homeomorphic to the inverse limit of an inverse sequence of dendrites with monotone bonding functions, which is by [20, Theorem 10.36, p. 180] a dendrite, and therefore ]im{[0,1],f}£=1
is a dendrite, too.
More specifically we prove that 1im{[0,1],f}£=1 is homeomorphic to
C),	1
1im{Dn,fn}n=1, where fn : Dn+1 ^ Dn is the mapping defined in Definition 3.8 and that each fn is monotone.

(Ö
CD
■ i-H
u
CD CO
For fixed x = (x1; x2, x3,..., xm, «n+i), xm = and fixed k, let
m> ""«,+1/ > ,Ajm
Bfc(x) = {(Xi,X2,X3, . . . ,Xm,«»+1^ °°),t G [ß„+1, 6„+i]}. Then each
o
IN
S(x)= U B(x),
is the star with the center x and beams (x), k = 1, 2, 3,... Using Remark 3.6 we see that
1. /n 1(x) = {x} for each x G Dn \ Cl(Dn+1 \ Dn), and
2. fn i(x) is the star S(x) for each x £ Dn if Cl(Dn+i \ Dn).
Therefore fn : Dn+i ^ Dn is monotone for each n and by [20, Theorem 10.36, p. 180]
£	]^{Dn, /„}~i
is a dendrite.
Next we show that by
o
a homeomorphism
£
is defined.
CO
F(x1,x2,x3,...) = lim x.
n
n^-oo
F : WD,/n}~1 ^ M0,1],/}n=1
1. First we show that F : l^m{Dn, /n}n=1 ^ Jm{[0,1],/}n=1 is a well defined function. Take any point (x1,x2,x3,...) in lm{Dn, /n}n=1 C n=1Di. If there is a positive integer n0, such that xn = xn0 for each n > n0, then lim xn = xn0 and xn0 G Dn0 C hm{[0,1],/}n=1. There-
n^ oo	l
fore F(x1, x2, x3,...) G lm{[0,1], /}n=1. If there is no such n0, then let i1 < i2 < i3 < ... be the sequence of all such integers that xin = xin+1
for each n. Then xin + 1 = xin + 1 G /in (xin ), where /in (xin ) is the star S(xin) C Din+1 with center xin. Therefore xin is of the form
xin = (y1,y2,y3,...,ym„ ,ain+1). Similarly, xin+1 is of the form
a
CD	o
xin+l = (z1, z2, z3, . . . , zmn+i, «m+2).

Since xin+1 G S(xin), it follows that mn < mn+1 and y = z for each i = 1,..., mn. From mn < mn+1 for each n, it follows that mn > n for each n. Therefore D(xin, xin+1) < ^sn < 2n. It follows that the sequence |xn}^=1 is a Cauchy sequence in Cl (UDn) and hence by Theorem 3.13 it converges to a point in Hm{[0,1], f }^=1.
o
CSI
CSI

CO CD
$H
CD CO
2. We show that F is continuous.
Take any x = (x1, x2, x3,...) G lim{Dn, fn}^=1 and any e > 0. Choose
ö
a positive integer k (given by Lemma 3.19), such that
diam( U (/k o ffc+1 o • • • o /n) 1(p)) < e
n>fc
for each p G Dk.
Let B = {z G W[0,1], /}~=1 | d(z, F(x)) < e} and let
vo
V = Pfc-1(B n Dfc),
where Pk : lim{Dn, fn}n=1 ^ Dk is the projection map to the k-th factor. Since B n Dk is open in Dk, V is open in llm{Dn, fn}n=1. Since x G lim{Dra, fn}n=1 and xk G Dk, it follows from the definition of F that F(x) G Cl(U„>fc(fk ◦ fk+1 ◦ ... ◦ fn)-1(xfc)). From the definition of functions f it follows that xfc G Cl(Un>k(fk ◦ fk+1 ◦... ◦ fn)-1(xk)). Therefore d(xk, F(x)) < e, hence xk G B. It follows that xk G B n Dk and thus x G V. Let y = (y1,y2,y3,...) G V .It follows that yk G B, and therefore d(yk,F(x)) < e. Since F(y),yk G Cl(Un>k(fk ◦ fk+1 ◦ ... ◦ fn)-1(yk)), it follows that d(yk, F(y)) < e. Hence,
d(F(x), F(y)) < d(F(x),yk) + d(yk,F(y)) < 2e. Therefore F is continuous. 3. We show that F is a surjection. Let
y =(y1,y2,y3,...) G ^nflO,1],f}n=1 be arbitrarily chosen. We define a sequence {xn}n=1, such that (a) for each n, xn G Dn,
Jh
(b)	for each n, /n(xn+1) = x„
CD
(c)	limn^ xn = y.
1—1
o
IN
ft
a
vD
0
a
o
1
00 ^
CO CO
If y G Dn for each n, then by Remark 3.14 y is of the form y =
(a»]1, aj22 > a»!, • • •), where for each I, it holds that ai£ < ai£+1 < bi£ and that k is a positive integer. In this case we define
xn
{n^2 n^3	^
(ai] , ai2 , ai3 , • • • , aim , aim + l )
where ^ < n for each I = 1, 2, 3, • • •, m, and > n. If y G for some m, then define xn = y for n > m and xn = (/n o • • • o /m-1)(y) for n < m.
Obviously the sequence {xn}n= satisfies (a), (b) and (c) and therefore F(xi,x2,x3, • • •) = y.
4. Finally we show that F is an injection. Let x = (x1 , x2,x3, • • •) and y = (yi, y2, y3, • • •) be any points in l^im{Dra, /ra}n=1, such that x = y. Let k be a positive integer such that xk = yk. Since xk ,yk G , it follows that
xfc
(all,
and
a92 a93 ai2 , ai3 ,

yfc = (al1 ,al2, a-, • • •, a£ , ),
where ¿1,¿2, • • •,¿j,I1,12, • • •,< k, t G (a^, bj] and s G (aim, &im], by Remark 3.6 (1). Let q = +q2+q3+• • •+9j, r = r1 +r2+r3+• • •+rm. Assume that q < r. Also, let n be the smallest integer such that Pn(xfc) = p„(yfc). If n < q then for each Z1 G Cl(Ui>k(/ o /+1 o • • • o /i)-1(xfc)) and each Z2 G Cl(U i>k (/ o /+1 o ••• o /i)-1(yfc)) by Lemma 3.18 it follows that pn(z1) = pn(xk) and pn(z2) = pn(yk), and therefore
D(Z1,Z2) >
d(Pn (xfc ),Pn(yfc ))
CO CD
$H
CD CO
$H
a
CD Jh
Since
and
F (x) G Cl^(/fc o /fc+1 o ••• o /i)-1(xfc))
i>fc
F (y) G Cl^(/fc o /fc+1 o ••• o /i)-1(yfc))
i>fc
it follows that F(x) = F(y). If n > q, then yk is of the form
t q1 9
yfc = (< > ai
92 aq3
i2 , ai3 ,

, aj, a
j+1 -
j+2 -
rj+3 lj+3 '
m
since r > q.
2
n
a
a

CM
We consider several cases.
o
Case 1. If p > 1, then n = q + 1, since pq+1(xk) = t and pq+1(yk) aj.,., and by Lemma 3.18 p„(F(x)) = p„(xfc) = t = aj. = p„(yfc) pn(F(y)), hence F(x) = F(y).
IN CM
Case 2. If p + rj+1 + rj+2 + rj+3 + ... + rm = 0, then n = q +1 and by Lemma 3.18 p„(F(x)) = p„(xfc) = t = s = p„(yfc) = p„(F(y)), hence F (x) = F (y).
£
Case 3. If p = 0 and rj+1 + rj+2 + rj+3 + ... + rm > 0 and if there is a positive integer i < k such that t = aj, then F(x) = xk and n < r + 1, and it follows that p„(F(x)) = p„(xfc) = p„(yfc) = p„(F(y)), where the last equality follows by Lemma 3.18.
Case 4. If p = 0 and rj+1 + rj+2 + rj+3 + ... + rm > 0 and if there is a positive integer i > k such that t = aj, then n = q + 1 since pg+1(yfc) = aj+1 and j+1 < k, while pq+1(xfc) = t = aj, i > k. Therefore pn(F(x)) = p„(xfc) = aj = a,+i = p„(yfc) = p„(F(y)), by Lemma 3.18.
Case 5. If p = 0 and rj+1 + rj+2 + rj+3 + ... + rm > 0 and if t = aj for each positive integer i, then F (x) = xk and n = q + 1 <r + 1, and we continue as in Case 3.
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CM CM
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Since F : ^im{Kra, fnjn=1 ^ ^!—{[0,1], /jn=1 is a continuous bijection from a compact space onto a metric space, it is by [19, Theorem 5.6, p. 167] a homeomorphism.	□
4 Wazewski's universal dendrite as an inverse limit with one bonding function
The following example shows that the conditions of Theorem 3.20 are not sufficient to guaranty that the corresponding inverse limit is homeomorphic to Wazewski's universal dendrite.
Jh
Example 4.1. Let for each positive integer n, (an, bn) = (1 — , 1 — 22i+i). By Theorem 3.20, lim{[0,1],/(an,bn
an+1 for each positive integer n, using Lemma 4.4 and Remark 3.6 (3), we see that x G ljm{[0,1], /(an bn)°° jk=1 is a ramification point if and only if there is a positive integer m, such that x = (a°°). Therefore the set of all ramification points is not dense in Hm{[0,1],/(«n,bn)^_ijk=1. Hence, by
U
cu
CM
i-H
o
CM
CD U
CU
Theorem 2.2, lim{[0,1],/(an,bn)
}°=i is not homeomorphic to Wazewski's
i °
)n = 1
universal dendrite.
In Theorem 4.5 we show that with the additional condition that the CM set {an | n = 1, 2, 3,...} is dense in [0,1], it follows that the inverse limit li—{[0,1], /}°=i is homeomorphic to Wazewski's universal dendrite. In Theorem 4.6 we show that in fact this additional condition characterizes inverse limits lm{[0,1], /}°=i that are homeomorphic to Wazewski's universal dendrite.
CD
First we prove the following lemmas.
Lemma 4.2. Let / : [0,1] ^ 2[0'i] be any comb function with respect to an admissible sequence {(an, bn)}n=i. Let
oo
y G lm{[0,1],/}°=i \U D
n
n=i
L\{y}c U D
n
n=i
be arbitrarily chosen. Then for each x G (Jn=i Dn the uniquely determined arc L from x to y satisfies the condition
o
Proof. By Remark 3.14 y is of the form y = (aj^l1, a^2 , a^3,...), where a^ < a^ < b^ for each l. We use the same sequence {xn}n=i as in the proof of surjectivity of F in the proof of Theorem 3.20, i.e.
CO
xn = (a^11 ^ ,ak33 ,...,ait:, a^) G Dn,
where i^ < n for each l = 1, 2, 3, ...,m, and im+i > n. Since Dn+i is a dendrite, there is a unique arc [xn,xn+i] from xn to xn+i in Dn+i if xn = xn+i. If xn = xn+i let [xn,xn+i] denote {xn}. Then A = (Jn=i[xn,xn+i] U {y} is an arc from xi to y, since [xn,xn+i] \ {xn} G Dn+i \ Dn and since lim
xn — y, as shown in the abovementioned proof of Theorem 3.20.
Jh
Obviously
oo
A\{y}c U Dn.
n=i
Next, take the unique arc B from xi to x in (Jn=i Dn (the existence of such an arc follows from the fact that there is an integer m such that xi, x G
Dm). Then Cl((A\B)U(B\A)) is an arc from x to y in W[0,1], /(ai)6i)°=1 }k=i
i-H
o
VD
O
CO CD
u
Since Hm{[0,1], /(ai,fei)==1}^ is a dendrite, it follows that Cl((A \ B) U (B \ A)) = L. Obviously
n=1
□
L \ {y} = Cl((A \ B) U (B \ A)) \ {y} C y Dn.
CN ctf
Lemma 4.3. Let f : [0,1] ^ 2[0'1] be any comb function with respect to an admissible sequence {(an, bn)}n=1. Then each
CD
y G Jim{[0,1],f}r=i \[j Dn
n=1
is an endpoint of lim{[0,1], f}(and hence it is not a ramification point)
Proof. Assuming that y is not an endpoint, using Lemma 4.2, one easily gets a simple closed curve in 1im{[0,1], f}f=1. This contradicts the fact that llm{[0,1], f}fc=1 is a dendrite by Theorem 3.20.	□
Lemma 4.4. Let f : [0,1] ^ 2[0'1] be any comb function with respect to an admissible sequence {(an, bn)}n=1. Let x G llm{[0,1], f}fc=1. The following statements are equivalent.
00
1.	x is a ramification point in lim{[0,1], f}£=1.
2.	x is a ramification point in Dn for some positive integer n.
Proof. It is obvious that if there is a positive integer n, such that x is a ramification point in Dn, then x is a ramification point in 1ini{[0,1], f}£=1 (since Dn Ç 11m{[0,1],f}£=1). Suppose that x is a ramification point in 11m{[0,1], f}fc=1. Since no point of
1m{[0,1],f} r=1 \Q Dn
n=1
is a ramification point in llm{[0,1],f}C=1, by Lemma 4.3, it follows that x G Dn0 for some positive integer n0. Let [x,xi], i = 1, 2, 3, be any three
arcs in Hin{[0,1],f}£=1, such that [x,x1] U [x,x2] U [x,x3] is a simple triod.
00
1
Without loss of generality we may assume that xi G Un=1 Dn, i = 1, 2, 3, since if xi G U^=1 Dn, we may replace [x,xi] by [x,yi], where yi G (x,xi), by Lemma 4.2. For each i = 1, 2, 3 let ni be a positive integer such that xi G Dni. Let n = max{n0, n1, n2, n3}. Obviously[x, x1] U [x, x2] U [x, x3] is a
CD
simple triod in Dn and therefore x is a ramification point in Dn.	□
DC
Theorem 4.5. Let f : [0,1] ^ 2[0'1] be any comb function with respect to an admissible sequence {(an, bn)}n=1 such that the set {an | n = 1, 2, 3,...} is dense in [0,1]. Then lim{[0,1], f }0=1 is homeomorphic to Wazewski's universal dendrite.
o
CM
O
CM CM
Proof. By Theorem 3.20, D = ljm{[0,1], f }k=1 is a dendrite. We show that the set of ramification points of D is dense in D and that each ramification point of D is of infinite order in D, and therefore by Theorem 2.2 D is
homeomorphic to Wazewski's universal dendrite.
Let y = (y^y2,y3,...) G D be arbitrarily chosen, such that y is not a ramification point. We show that there is a sequence of ramification points {zn}n=1 in D, such that lim zn = y.
n^oo
If y G Dn for some positive integer n, then by Remark 3.6 (1), (3) (taking into account that by Lemma 4.4 y is not a ramification point in Dl for each I) there are a positive integer m and a real number t G [0,1] \{a?, a2, a3,...}, such that
y = (y1,y2,y3,...,ym_1,t 0),
where ym_1 = ak for some k < n, and t G (ak, bk]. Since the set {an | n = 1, 2, 3,...} is dense in [0,1], there is a strictly increasing sequence {in}n= of positive integers, such that lim ain = t and ain G (ak, bk]. Therefore
n^ oo
{(y1,y2 ,y3,...,ym_1,ain )}n=1
is a sequence of ramification points in D, which converges to y. 0 D
n=1 Dn
If y G D \ Un=? Dn, then by Remark 3.14
£
y = « ,ak2 ,ak33,...), 1 2 3
where for each I it holds that kl > 0 and ai£ < ai£+1 < bi£. Then the sequence {zn}n=1, where
Zn = (akl ,ak22 ,ak33,..., akn-, )
for each n, is a sequence of ramification points in D, which converges to y.
Next we show that each of the ramification points is of infinite order in D. Let x G D be any ramification point. Then by Lemma 4.4 and Remark 3.6 (3) there are positive integers m and j, such that pk(x) = aj for each e positive integer k > m. Without loss of generality we may assume that pk (x) = aj for each k < m. Since
x G fj_11(x) C D
and fj__11(x) is a star with the center x by Lemma 3.10, it follows that x is of infinite order in D.	□
o
CM
Theorem 4.6. Let f : [0,1] ^ 2[0'1] be any comb function with respect to an admissible sequence {(an, bn)}^c=1. Then 1i—{[0,1],f }k=1 is homeomorphic to Wazewski's universal dendrite if and only if the set {an | n = 1, 2, 3,...} is dense in [0,1].
CM
Proof. Taking Theorem 4.5 into account it suffices to prove that if the set {an | n = 1, 2, 3,...} is not dense in [0,1], then 1i—{[0,1],f }£=1 is not homeomorphic to Wazewski's universal dendrite. If there is an interval J = (aj, ak) C [0,1] containing no an, let t = + "fe and 6 = "fef . For any
ramification point x of 1im{[0,1],f}^=1 D(x, (t~)) > d(plfx)'t} > 6, since
_1 JS	J J	f
p1(x) = an for some n. Therefore the open ball in ]im{[0,1], f }£=1 centered at (t^) with the radius 6 contains no ramification points and hence by Theorem 2.2 ]im{[0,1], f }£=1 is not homeomorphic to Wazewski's universal dendrite.	□
£
Theorem 4.7. There is a comb function f such that ]im{[0,1],f}£=1 is , 1
homeomorphic to Wazewski's universal dendrite.
Proof. Let {an | n £ N} be any dense subset of (0,1). Inductively we define sequence {bn}n=1 in such a way that {(an, bn)}n=1 would be admissible which would by Theorem 4.5 guaranty that ]im{[0,1],f}£=1 is homeomorphic to Wazewski's universal dendrite. For each positive integer n, let
CM
1
bn = 2 (an + min{1, a* | i < n, a* > a„}).
CO
First we show that limn^^(bn — an) = 0. Let £ > 0 be arbitrary; choose a positive integer k such that k < £. For each j < k choose ij, such that aj £ (^IT, I), and let n0 = max{ij | j = 1, 2, 3,..., k}. For any n > n0 let a < b be two consecutive elements of the set {0,1, a*. | j = 1, 2, 3,..., k}
such that an £ (a, b). Then bn — an < ^^ — an = ^^ < ^ < £.
Since for each positive integer n for each m > n it holds that if am < an, then bm < 1 (am + an) < an, it follows that the sequence {(an,bn)}n=1 is
admissible.	□
Acknowledgements
This work was supported in part by the Slovenian Research Agency, under
U
a
CD
Grants P1-0285 and P1-0297.
References
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Iztok Banic
(1)	Faculty of Natural Sciences and Mathematics, University of Maribor,
Koroska 160, Maribor 2000, Slovenia
(2)	Institute of Mathematics, Physics and Mechanics,
Jadranska 19, Ljubljana 1000, Slovenia
E-mail: iztok.banic@uni-mb.si
O
CM
Matevž Crepnjak
(1)	Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia
(2)	Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova 17, Maribor 2000, Slovenia
E-mail: matevz.crepnjak@uni-mb.si
£
Matej Merhar
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia E-mail: matej.merhar@uni-mb.si
i—l
Uroš Milutinovic
(1)	Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia
(2)	Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana 1000, Slovenia
E-mail: uros.milutinovic@uni-mb.si
CM
Tina Sovič
2 Faculty of Civil Engineering, University of Maribor, CM
Smetanova 17, Maribor 2000, Slovenia E-mail: tina.sovic1@uni-mb.si
CO CD
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