On inverse limits with set-valued functions on graphs, dimensionally stepwise spaces and ANRs

Topology and its Applications 226 (2017) 16–30

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Topology and its Applications www.elsevier.com/locate/topol

On inverse limits with set-valued functions on graphs, dimensionally stepwise spaces and ANRs Masatoshi Hiraki, Hisao Kato Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan

a r t i c l e

i n f o

Article history: Received 5 April 2017 Accepted 19 April 2017 Available online 26 April 2017 MSC: primary 54F15, 54F45, 54C56 secondary 54C60, 54B20

a b s t r a c t A space X is a dimensionally stepwise space if dim X < ∞ and for any 1 ≤ m ≤ dim X there is an open set Um of X such that dim Um = m. In this paper, for given inverse sequence {Xi , fi,i+1 }∞ i=1 of compacta with upper semi-continuous set-valued ˜ ˜ functions, we introduce new indexes I({X i , fi,i+1 }) and W ({Xi , fi,i+1 }), and by use of the indexes we investigate topological structures of inverse limits of graphs with upper semi-continuous set-valued functions. Especially, we prove the following theorems. © 2017 Elsevier B.V. All rights reserved.

Keywords: Continua Inverse limits Inverse limits with set-valued functions Dimension Shape Cell-like FAR ANR Dendrite

Theorem 0.1. Let G be a graph and let f : G → 2G be an upper semi-continuous function. If the inverse limit P = ← lim {G, f } with the single upper semi-continuous bonding function f is a polyhedron, then P is a −− dimensionally stepwise space, i.e., for any natural number i with 1 ≤ i ≤ dim P , P has a free i-simplex. Theorem 0.2. If f : G → 2G is an upper semi-continuous function on a graph G such that dim D1 (f −1 ) ≤ 0 ˜ ({G, f }) < ∞, then the inverse limit lim{G, f } with the single upper semi-continuous bonding function and W ←−− f is a dimensionally stepwise space. Theorem 0.3. Suppose that Ii (i ∈ N) is a sequence of the unit interval I = [0, 1] and Ki is a finite simplicial complex in Ii × Ii+1 satisfying that for any x ∈ Ii+1 , (Ii × {x}) ∩ |Ki | = ∅ and for any y ∈ Ii , ({y} × Ii+1 ) ∩ |Ki | is a nonempty connected set (=a closed interval). Let fi,i+1 : Ii+1 → 2Ii be the surjective E-mail addresses: [email protected] (M. Hiraki), [email protected] (H. Kato). http://dx.doi.org/10.1016/j.topol.2017.04.022 0166-8641/© 2017 Elsevier B.V. All rights reserved.

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upper semi-continuous function defined by G(fi,i+1 ) = |Ki |. Then ← lim {I , f } is an AR. Moreover, if −− i i,i+1 ˜ i , fi,i+1 }) = 0, then lim{Ii , fi,i+1 } is a dendrite. dim |Ki | ≤ 1 (i ∈ N) and I({I ←−− 1. Introduction Inverse limits with bonding maps have played very important roles in the development of topology and topological dynamical systems. In fact, every complicated compactum can be represented by an inverse limit of finite polyhedra and simple bonding maps and conversely, inverse limits with simple bonding maps are very useful to construct complicated spaces. In 2004, Mahavier started studying inverse limits with set-valued functions as inverse limits with closed subsets of the unit square [0, 1] × [0, 1] ([13]). Since then, several topological properties of inverse limits of compacta with upper semi-continuous set-valued functions have been studied by many authors (see [1,3, 6–11,13–15]). In [8,10], Ingram and Mahavier discussed several results concerning connectedness, indecomposability and dimension of such inverse limits. Also, they investigated many interesting examples of such inverse limits of set-valued functions from the unit interval I = [0, 1] to I. Such examples give us important suggestions on understanding of inverse limits. Main ideas of the present paper follow from such examples. The study of such inverse limits has developed into one of rich topics of continuum theory. Note that there are many differences between the theory of inverse limits with mappings (=single valued functions) and the theory with set-valued functions. Banič, Nall and Ingram studied topological dimension of such inverse limits (see also [1,8–10,14]). It is well-known that inverse limits of sequences of single-valued continuous functions (=mappings) have dimension bounded by the dimensions of the factor spaces. In [14], Nall proved that inverse limits of sequences of upper semi-continuous set-valued functions with 0-dimensional values have dimension bounded by the dimensions of the factor spaces. In [3], Charatonik and Roe investigated trivial shape properties of such inverse limits (see also [11]). The following general problem remains open. Problem 1.1. What compactum can be obtained as an inverse limit with a sequence of upper semi-continuous bonding functions on graphs, especially [0, 1]? Especially, we have the following problem. Problem 1.2. What compactum can be obtained as an inverse limit with a single upper semi-continuous bonding function on a graph, especially [0, 1]? In [7], Illanes proved that a simple closed curve is not an inverse limit on [0, 1] with a single uppersemicontinuous bonding function. In [15], Nall proved that the arc is the only finite graph that is an inverse limit on [0, 1] with a single upper-semicontinuous bonding function. Also, in [14] Nall showed that an inverse limit with a single upper semi-continuous bonding function on [0, 1] cannot be an n-cell (n ≥ 2). In [11], to evaluate the dimension of the inverse limit of given inverse sequence {Xi , fi,i+1 }∞ i=1 of compacta with upper ˜ semi-continuous set-valued functions, we defined an index J({X i , fi,i+1 }). In this paper, we will introduce ˜ ˜ other indexes I({X i , fi,i+1 }) and W ({Xi , fi,i+1 }), and by use of the indexes we investigate the topological structures of inverse limits of graphs with upper semi-continuous set-valued functions. We give some partial answers to the above two problems. 2. Definitions and notations In this paper, we assume that all spaces are separable metric spaces and maps (=mappings) are continuous functions. We use dim X for the topological dimension of a space X. A space X is a dimensionally stepwise space if dim X < ∞ and for any 1 ≤ m ≤ dim X there is an open set Um of X such that dim Um = m.

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Note that all 0-dimensional spaces and all 1-dimensional spaces are dimensionally stepwise spaces. A space X is a compactum if X is a compact metric space. A space X is a continuum if X is a connected compactum. A continuum H is a Cantor n-dimensional manifold provided that if S is any separator of H, then dim S ≥ n − 1. Note that any n-dimensional compactum contains a Cantor n-dimensional manifold (see [5, p. 73, Cantor-manifold theorem (1.9.9)]). A space G is a graph if G is a 1-dimensional compact connected polyhedron. Tree means a graph containing no simple closed curve. A subspace J of G is a free arc in G if J is homeomorphic to the unit interval I = [0, 1] and J − {e, e } is an open set of G, where e and e are the two end points of J. Let X and Y be compacta. Let 2X be the collection of all nonempty closed subsets of X and C(X) the collection of all nonempty subcontinua of X. A function f : X → 2Y is said to be upper semi-continuous if for any point x ∈ X and any open neighborhood V of f (x), there is an open neighborhood U of x such that if x ∈ U , then f (x ) ⊂ V . A function f : X → 2Y is said to be surjective if f (X) = Y , where f (A) = ∪{f (x)| x ∈ A} for a subset A of X. If f : X → 2Y , g : Y → 2Z are set-valued functions, then we define the composition gf : X → 2Z by gf (x) = g(f (x))(= ∪{g(y)| y ∈ f (x)}) for x ∈ X. For a function f : X → 2Y , we put D1 (f ) = {x ∈ X| dim f (x) ≥ 1}, D1 (f −1 ) = {y ∈ Y | dim f −1 (y) ≥ 1}, where f −1 (B) = {x ∈ X| f (x) ∩ B = φ} for a subset B of Y . Let N be the set of natural numbers. Let Xi (i ∈ N) be a sequence of compacta and let fi,i+1 : Xi+1 → 2Xi be an upper semi-continuous function for each i ∈ N. The inverse limit of the inverse sequence {Xi , fi,i+1 }∞ i=1 is the space lim {Xi , fi,i+1 } = ←−−

{(xi )∞ i=1

| xi ∈ fi,i+1 (xi+1 ) for each i ∈ N} ⊂

∞ 

Xi

i=1

∞ which has the topology inherited as a subspace of the product space i=1 Xi . For i ≤ j, we define fi,j : Xj → 2Xi by fi,i = id, fi,j = fi,i+1 fi+1,i+2 · · · fj−1,j : Xj → 2Xi (i < j). In particular, if f : X → 2X is an upper semi-continuous function, we consider the inverse sequence {X, f } = {Xi , fi,i+1 }, where Xi = X, fi,i+1 = f (i ∈ N). We put lim {X, f } = {(xi )∞ i=1 | xi ∈ f (xi+1 ) for each i ∈ N}. ←−− Let {Xi , fi,i+1 }∞ i=1 be an inverse sequence with set-valued functions. For each m ≤ n, we put G(f ; m, m + 1, · · · , n) = {(xi )ni=m ∈

n 

Xi | xi ∈ fi,i+1 (xi+1 ) for each m ≤ i ≤ n − 1}.

i=m

In particular, G(f1,2 ) = G(f ; 1, 2) = {(x1 , x2 ) ∈ X1 × X2 | x1 ∈ f1,2 (x2 )} is the graph of f1,2 : X2 → 2X1 . Let π[m,n] : ← lim {Xi , fi,i+1 } → G(f ; m, m + 1, ..., n) be the natural projection −− defined by π[m,n] (x1 , x2 , ...xm , ..., xn , xn+1 , ...) = (xm , ..., xn ). Let y ∈ Xn and x ∈ Xn (n ≤ n ). In [11], we consider the following conditions:

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y ←x : y ∈ fn,n (x) x
: x ∈ D1 (fn−1  ,n +1 ) y : n ≥ 2 and y ∈ D1 (fn−1,n ) Also, let x ∈ Xm and y ∈ Xm (m + 2 ≤ m ). We consider the following condition: −1 x ≺y : y ∈ D1 (fm −1,m ) and dim[fm,m  −1 (x) ∩ fm −1,m (y)] ≥ 1

In particular, −1 −1 x 3y : m = m + 2, x ∈ D1 (fm,m+1 ), y ∈ D1 (fm+1,m+2 ) and dim[fm,m+1 (x) ∩ fm+1,m+2 (y)] ≥ 1 −1 For each xn ∈ Xn with xn ∈ D1 (fn,n+1 ), we consider the following sequence:

ym1 ≺ ym2 ≺ ym3 ≺ · · · ≺ ymk−1 ≺ ymk ← xn
, where 2 ≤ m1 , mk ≤ n, mi + 2 ≤ mi+1 (i = 1, 2, ..., k − 1) and ymi ∈ Xmi (i = 1, 2, ..., k). In this case, we say that the sequence (ym1 , ym2 , · · · , ymk , xn ) is an expand-contract sequence in {Xi , fi,i+1 }∞ i=1 with length k. For any expand-contract sequence S : ym1 ≺ ym2 ≺ ym3 ≺ · · · ≺ ymk−1 ≺ ymk ← xn
, we put d(S) =

k i=1

˜ dim fmi −1,mi (ymi ). We define the index J({X i , fi,i+1 }) as follows.

∞ ˜ J({X i , fi,i+1 }) = sup{d(S) | S is an expand-contract sequence in {Xi , fi }i=1 }.

˜ If each Xi is 1-dimensional, then J({X i , fi,i+1 }) is the maximal length of all expand-contract sequences in ∞ ˜ {Xi , fi,i+1 }i=1 . If there is no expand-contract sequence in {Xi , fi,i+1 }∞ i=1 , we put J({Xi , fi,i+1 }) = 0 (see [11]). In [11], we proved the following theorems. Theorem 2.1. ([11, Theorem 4.1]) Let Xi (i ∈ N) be a sequence of compacta and let fi,i+1 : Xi+1 → 2Xi be an upper semi-continuous function for each i ∈ N. Suppose that dim D1 (fi,i+1 ) ≤ 0 (i ∈ N). Then ˜ dim lim {Xi , fi,i+1 } ≤ J({X i , fi,i+1 }) + sup{dim Xi | i ∈ N}. ←−− Theorem 2.2. ([11, Theorem 3.8]) Let Xi (i ∈ N) be a sequence of 1-dimensional compacta and let fi,i+1 : Xi+1 → 2Xi be a surjective upper semi-continuous function for each i ∈ N. Suppose that each i ≥ 2, Zi is a 0-dimensional closed subset of Xi such that fi,i+1 |Xi+1 − Zi+1 : (Xi+1 − Zi+1 ) → Xi is a mapping for each ˜ x ∈ Xi+1 − Zi+1 and i ∈ N. If J({X i , fi,i+1 }) = k, then k ≤ dim ← lim {Xi , fi,i+1 } ≤ k + 1. −− Moreover, if there is an expand-contract sequence ym1 ≺ ym2 ≺ ym3 ≺ · · · ≺ ymk−1 ≺ ymk ← xn
−1 ˜ in {Xi , fi,i+1 } with length J({X i , fi,i+1 }) = k such that dim pn (xn ) > 0, then

dim lim {Xi , fi,i+1 } = k + 1, ←−− where pn : lim {Xi , fi,i+1 }i≥n → Xn is the projection defined by pn (xn , xn+1 , · · · ) = xn . ←−−

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∞ ˜ Now, we will define another index I({X i , fi,i+1 }) as follows. Let {Xi , fi,i+1 }i=1 be an inverse sequence  with set-valued functions. Also, let x ∈ Xm and y ∈ Xm (m +2 ≤ m ). We consider the following condition: −1 −1 x
y : x ∈ D1 (fm,m+1 ) and dim[fm,m+1 (x) ∩ fm+1,m (y)] ≥ 1

Note that x3y implies x ≺ y and x
 y. For each xn ∈ Xn with xn ∈ D1 (fn−1,n ), we consider the following sequence: xn ← ym1
 ym2
 ym3
 · · ·
 ymk−1
 ymk
where n ≤ m1 , mi + 2 ≤ mi+1 (i = 1, 2, ..., k − 1) and ymi ∈ Xmi (i = 1, 2, ..., k). In this case, we say that the sequence (xn , ym1 , ym2 , · · · , ymk ) is an inverse expand-contract sequence in {Xi , fi,i+1 }∞ i=1 with length k. Note that a sequence (xn , ym1 , ym2 , · · · , ymk ) is an inverse expand-contract sequence in the inverse sequence {Xi , fi,i+1 }∞ i=1 if and only if the sequence (ymk , ymk−1 , · · · , ym1 , xn ) is an expand-contract sequence in the −1 direct sequence {Xi , fi,i+1 }∞ i=1 . For any inverse expand-contract sequence S : xn ← ym1
 ym2
 ym3
 · · ·
 ymk−1
 ymk
we put d(S) =

k i=1

−1 ˜ dim fm (ymi ). We define the index I({X i , fi,i+1 }) as follows. i ,mi +1

˜ I({X i , fi,i+1 }) = sup{d(S) | S is an inverse expand-contract sequence in {Xi , fi,i+1 }}. ˜ If there is no inverse expand-contract sequence in {Xi , fi,i+1 }∞ i=1 , we put I({Xi , fi,i+1 }) = 0. Note that for X ˜ −1 ˜ any upper semi-continuous function f : X → 2 , J({X, f }) = I({X, f }). Also, we consider the following sequence S in {Xi , fi,i+1 }∞ i=1 . S : ym1
 ym2
 ym3
 · · ·
 ymk−1
 ymk
In this case, we say that the sequence {ymi | 1 ≤ i ≤ k} is a weak inverse expand-contract sequence in k −1 {Xi , fi,i+1 }∞ i=1 with length k. We put d(S) = i=1 dim fmi ,mi +1 (ymi ) and define the index ˜ ({Xi , fi,i+1 }) = sup{d(S) | S is a weak inverse expand-contract sequence in {Xi , fi,i+1 }}. W ˜ ˜ If each Xi is 1-dimensional, then I({X i , fi,i+1 }) (resp. W ({Xi , fi,i+1 })) is the maximal length of all (resp. ˜ ˜ weak) inverse expand-contract sequences in {Xi , fi,i+1 }. Note that I({X i , fi,i+1 }) ≤ W ({Xi , fi,i+1 }). In ˜ ˜ ˜ general, the index I({X, f }) is not equal to the index J({X, f }), and I({X, f }) is not equal to the index ˜ W ({X, f }). See the next example. Example 1. Let C be a Cantor set in [1/2, 3/4] with 1/2, 3/4 ∈ C. Let f : I → 2I be the surjective upper semi-continuous function defined as follows: f ([0, 1/4]) = 0, f (1/4) = [0, 1/4], f ((1/4, 1/2)) = 1/4 and f |[1/2, 3/4] : [1/2, 3/4] → [1/4, 1/2] is a map with f (C) = [1/4, 1/2], f (3/4) = C, f ((3/4, 1)) = 3/4, f (1) = [3/4, 1]. Then 1/4 ← 1/4
 3/4
is a maximal inverse expand-contract sequence in {X, f }. Note that there is no x ∈ I such that 1/4 ≺ x ← 3/4
.

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Also, 1/4 ← 3/4
, 1/4 ← 1/4
˜ ˜ are maximal expand-contract sequences in {X, f }. Hence J({I, f }) = 1 < 2 = I({I, f }). Let g = f −1 : I → I ˜ ˜ ˜ ˜ 2 . Then I({I, g}) = J({I, f }) = 1 < 2 = I({I, f }) = J({I, g}). Consider the map h : J = [1/4, 1] → 2J ˜ ˜ ({J, h}). defined by h(x) = f (x) ∩ [1/4, 1]. Then I({J, h}) = 0 < 1 = W The following results are well-known. Theorem 2.3. (Hurewicz’s theorem [5, p. 242]) If f : X → Y is a closed mapping between separable metric spaces and there is k ≥ 0 such that dim f −1 (y) ≤ k for each y ∈ Y , then dim X ≤ dim Y + k. Theorem 2.4. (Vaˇinšteˇin’s theorem [5, p. 244]) Suppose that f : X → Y is a closed mapping between separable metric spaces. If dim Y ≤ n and dim Di (f −1 ) ≤ n − i for each i = 1, 2, ..., n + 1, then dim X ≤ n, where Di (f −1 ) = {y ∈ Y | dim f −1 (y) ≥ i}. The next result (Theorem 2.5) can be proved by symmetrical argument of the proof of Theorem 2.1 (see [11, Theorem 4.1]). For completeness, we give the proof. Theorem 2.5. Let Xi (i ∈ N) be a sequence of compacta and let fi,i+1 : Xi+1 → 2Xi be an upper semi−1 continuous function for each i ∈ N. Suppose that dim D1 (fi,i+1 ) ≤ 0 (i ∈ N). Then ˜ dim lim {Xi , fi,i+1 } ≤ I({X i , fi,i+1 }) + sup{dim Xi | i ∈ N}. ←−− ˜ Proof. We may assume that sup{dim Xi |i ∈ N} = m < ∞ and I({X i , fi,i+1 }) = k < ∞. We prove the following claim (*): (*) If Xi (i ∈ N) is a sequence of compacta and fi,i+1 : Xi+1 → 2Xi is an upper semi-continuous function −1 for each i ∈ N satisfying the condition dim D1 (fi,i+1 ) ≤ 0 (i ∈ N), then for any r ≥ 2 ˜ dim G(f ; 1, 2, ..., r) ≤ I({X i , fi,i+1 }) + sup{dim Xi | i ∈ N} = n. ˜ We proceed by induction on I({X i , fi,i+1 }) = k. First, we consider the case k = 0. ˜ Case (0): I({Xi , fi,i+1 }) = 0. We will prove that for any r ≥ 2, dim G(f ; 1, 2, ..., r) ≤ sup{dim Xi | i ∈ N}. We prove the case (0) by induction on r. Let r = 2. Consider the projection p1 : G(f ; 1, 2) = G(f1,2 ) → X1 . Note that p1 is a closed mapping and −1 dim D1 (p−1 1 ) = dim D1 (f1,2 ) ≤ 0.

We will apply Theorem 2.4 to the case f = p1 and n = max{dim X1 , dim X2 }. Then we see that dim G(f ; 1, 2) ≤ max{dim X1 , dim X2 } ≤ sup{dim Xi | i ∈ N}. Next we assume that for r (≥ 2) dim G(f ; 1, 2, ..., r) ≤ sup{dim Xi | i ∈ N}.

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We must prove that dim G(f ; 1, 2, ..., r, r + 1) ≤ sup{dim Xi | i ∈ N}. By the assumption, we may assume that dim G(f ; 1, ..., r) ≤ sup{dim Xi | i ∈ N}. Consider the projection q : G(f ; 1, 2, ..., r, r + 1) → G(f ; 1, ..., r) defined by q(x1 , x2 , ...., xr , xr+1 ) = (x1 , x2 , ...., xr ). Note that −1 D1 (q −1 ) = {(x1 , x2 , ...., xr ) ∈ G(f ; 1, ..., r)| xr ∈ D1 (fr,r+1 )} −1 ˜ and dim D1 (fr,r+1 ) ≤ 0. Since I({X i , fi,i+1 }) = 0, we see that the following condition holds; (3) if xr ∈ −1 D1 (fr,r+1 ) and xj ∈ Xj (2 ≤ j ≤ r) with xj ← xr , then xj ∈ / D1 (fj−1,j ), i.e., dim fj−1,j (xj ) = 0. For each 1 ≤ j ≤ r, we consider the space

˜ ; j, ..., r) = {(xj , ...., xr ) ∈ G(f ; j, ..., r)| xr ∈ D1 (f −1 )} G(f r,r+1 ˜ ; j, ..., r) → G(f ˜ ; j + 1, ..., r) defined by and the projection qj : G(f qj (xj , ....xr ) = (xj+1 , ...., xr ). ˜ ;j+ By the condition (3), qj is 0-dimensional, i.e., dim(qj )−1 (xj+1 , ...., xr ) ≤ 0 for each (xj+1 , ...., xr ) ∈ G(f 1, ..., r). For each j, we apply Theorem 2.3, we see −1 ˜ ; r − 1, r) = dim G(f ˜ ; r − 2, r) = 0 ≥ dim D(fr,r+1 ) = dim G(f

˜ ; 1, 2, · · · , r) = D1 (q −1 ). · · · = G(f Hence dim D1 (q −1 ) ≤ 0. Applying Theorem 2.4 for f = q and n = sup{dim Xi | i ∈ N}, we see that dim G(f ; 1, 2, ..., r, r + 1) ≤ sup{dim Xi | i ∈ N}. ˜ ˜ Case (k): I({X i , fi,i+1 }) = k (k ≥ 1). We assume that (*) is true for the case I({Yi , gi,i+1 }) ≤ k − 1, Yi i.e., if Yi (i ∈ N) is a sequence of compacta and gi,i+1 : Yi+1 → 2 is an upper semi-continuous function −1 ˜ i , gi,i+1 }) ≤ k − 1, then for any for each i ∈ N satisfying the condition dim D1 (gi,i+1 ) ≤ 0 (i ∈ N) and I({Y r∈N ˜ i , gi,i+1 }) + sup{dim Yi | i ∈ N}. dim G(g; 1, 2, ..., r) ≤ I({Y ˜ We will show that the claim (*) is true for the case I({X i , fi,i+1 }) = k. Let r = 2. As before, we see that ˜ dim G(f ; 1, 2) ≤ max{dim X1 , dim X2 } ≤ I({X i , fi,i+1 }) + sup{dim Xi | i ∈ N}. Now, we suppose that for r (≥ 2), the following is true:

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˜ dim G(f ; 1, 2, ..., r) ≤ I({X i , fi,i+1 }) + sup{dim Xi | i ∈ N}. We must show that ˜ dim G(f ; 1, 2, ..., r, r + 1) ≤ I({X i , gi,i+1 }) + sup{dim Xi | i ∈ N}. Consider the projection q : G(f ; 1, 2, ..., r, r + 1) → G(f ; 1, 2, ..., r). Recall that −1 D1 (q −1 ) = {(x1 , ...., xr ) ∈ G(f ; 1, 2, ..., r)| xr ∈ D1 (fr,r+1 )} −1 −1 and dim D1 (fr,r+1 ) ≤ 0. Fix z ∈ D1 (fr,r+1 ) ⊂ Xr . Consider the following inverse sequence {Yi , gi,i+1 }∞ i=1 , where Yi = fi,r (z) (i ≤ r), Yi = {z} (i ≥ r) and gi,i+1 : Yi+1 → 2Yi (i ≤ r − 1) is defined by −1 ˜ gi,i+1 (x) = fi,i+1 (x) and gi,i+1 (z) = {z} (i ≥ r). Since I({X i , fi,i+1 }) = k and z ∈ D1 (fr,r+1 ), we see −1 ∞ ˜ that I({Yi , gi,i+1 }i=2 ) ≤ k − dim fr,r+1 (z), and hence by the inductive assumption we see that for the fixed −1 point z ∈ D1 (fr,r+1 ),

dim {(x1 , ..., z) ∈ G(f ; 1, ..., r)} = dim G(g; 1, 2, ..., r) −1 (z)) + sup{dim Yi | i ∈ N} ≤ (k − dim fr,r+1 −1 ≤ (k − dim fr,r+1 (z)) + sup{dim Xi | i ∈ N}.

Recall that −1 Di (q −1 ) = {(x1 , ..., z) ∈ G(f ; 1, 2, ..., r) | dim fr,r+1 (z) ≥ i}.

For each i ≥ 1, consider the projection −1 −1 πr : Di (q −1 ) → {z ∈ Xr | dim fr,r+1 (z) ≥ i} ⊂ D1 (fr,r+1 )

defined by πr (x1 , ..., z) = z. By the Hurewicz’s theorem (Theorem 2.3), we see that for each i ≥ 1, dim Di (q −1 ) ≤ (k − i) + sup{dim Xi | i ∈ N}. We apply the Vaˇinšteˇin’s theorem (Theorem 2.4) to the case f = q and n = k + sup{dim Xi | i ∈ N} and we can conclude that ˜ dim G(f ; 1, 2, ..., r, r + 1) ≤ I({X i , fi,i+1 }) + sup{dim Xi | i ∈ N}. Hence (*) is true for any r ≥ 2. For any  > 0 there exists a sufficiently large r ∈ N such that the natural projection π[1,r] : {Xi , fi,i+1 } → G(f ; 1, 2, ..., r) is an -map. By [5], we can conclude lim ←−− ˜ dim lim {Xi , fi,i+1 } ≤ I({X i , fi,i+1 }) + sup{dim Xi | i ∈ N}. ←−− This completes the proof. 2 Combining Theorem 2.1 and Theorem 2.5, we have the following. Theorem 2.6. Let Xi (i ∈ N) be a sequence of compacta and let fi,i+1 : Xi+1 → 2Xi be an upper semi−1 ) ≤ 0 (i ∈ N). continuous function for each i ∈ N. Suppose that dim D1 (fi,i+1 ) ≤ 0 and dim D1 (fi,i+1 Then ˜ ˜ dim lim {Xi , fi,i+1 } ≤ min{I({X i , fi,i+1 }), J({Xi , fi,i+1 })} + sup{dim Xi | i ∈ N}. ←−−

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3. Dimensionally stepwise spaces and inverse limits with set-valued functions In this section, we will give a partial answer to Problem 1.2. In [14], Nall proved that an inverse limit with a single upper semi-continuous bonding function on [0, 1] cannot be an n-cell (n ≥ 2). We will use an idea of Nall in the proof of [14, Lemma 3.1 and Theorem 3.2]. We need the following. Proposition 3.1. Let G be a graph and let f : G → 2G be an upper semi-continuous function. If there is a point x ∈ G such that dim π1−1 (x) = m, then there is a free arc J of G such that dim π1−1 (Int(J)) = m lim {G, f } such that and dim π1−1 (z) ≥ m − 1 for each z ∈ J. In particular, there is an open set Um of ← −− dim Um = m. lim {G, f } → lim {G, f } be the shift map defined by σf (x1 , x2 , x3 , ..., ) = (x2 , x3 , ..., ). Note Proof. Let σf : ← −− ←−− that if H is a subset of ← lim {G, f } such that π[1,k] (H) is degenerate, then (σf )i |H : H → lim {G, f } is −− ←−− injective for 1 ≤ i ≤ k (see the proof of Nall in [8, Theorem 5.5]). Let x = x1 be a point of G such that dim π1−1 (x1 ) = m. Then we can choose a Cantor m-dimensional manifold H in π1−1 (x1 ). Let k ∈ N such that πk (H) is non degenerate and πi (H) is degenerate for each 1 ≤ i < k. Let J be a free arc of G with J ⊂ Int(πk (H)). Note that there do not exist two points z, z  ∈ J (z = z  ) such that dim π1−1 ({z, z  }) ≤ m − 2 because that π1−1 ({z, z  }) separates the continuum (k−1) H  = σf (H) which is homeomorphic to the Cantor m-dimensional manifold H. Hence we can choose a small free arc J such that dim π1−1 (z) ≥ m − 1 for each z ∈ J. Put K = {(xi )∞ {G, f }| xi = πi (H) for 1 ≤ i < k and xk ∈ J}. i=1 ∈ lim ←−− Then K ⊂ π1−1 (x1 ) and K contains a nonempty open set of H, hence K is m-dimensional. Let K  = π1−1 (J) (= {(yi )∞ lim {G, f }| y1 ∈ J}) (= σf i=1 ∈ ← −−

(k−1)

(K)).

Since K and K  is homeomorphic, K  is m-dimensional. Put Um = π1−1 (Int(J)) ⊂ K  . Note that Um contains a nonempty open set of f˜(k−1) (H). Then dim Um = m. 2 Let P be a polyhedron. A simplex Δ in P is a free simplex of P if the interior Δ − ∂Δ of Δ is an open set of P , i.e., Δ is not a face of any other simplex. A known example of something that could be obtained as such an inverse limit with single upper semi-continuous set valued function on the unit interval I is a 2-cell with an arc attached. So, someone might think that it is possible to construct a 3-cell with an attached 2-cell as such an inverse limit. But, the next result implies that it is impossible. In fact, we show that if an inverse limit with single upper semi-continuous set valued function on a graph is a polyhedron, then that polyhedron must contain a free i-simplex for each i between from 1 up to the dimension of the polyhedron. Theorem 3.2. Let G be a graph and let f : G → 2G be an upper semi-continuous function. If the inverse limit ← lim {G, f } is homeomorphic to a polyhedron P , then P is a dimensionally stepwise space, i.e., for any −− natural number i with 1 ≤ i ≤ dim P , P has a free i-simplex. Proof. Since P is a polyhedron, the following condition (∗i ) is true: (∗i ) If U is an open set of P with dim U = i (i ≥ 1), then U can not contain uncountable mutually disjoint i-dimensional subsets. Let dim P = m. We may assume that m ≥ 2. Consider the map π1 : P → G. By the Hurewicz’s theorem (see [5, p. 242]), we can find a point y ∈ G such that dim π1−1 (y) ≥ m − 1. If dim π1−1 (y) = m, by Proposition 3.1 there is a free arc J1 of G such that dim π1−1 (Int(J)) = m and dim π1−1 (z) ≥ m − 1 for each z ∈ J1 . By the condition (∗m ), we can find a point y1 ∈ J1 such that dim π1−1 (y1 ) = m − 1. Also, by Proposition 3.1 we can find a free arc J2 such that dim π1−1 (Int(J2 )) = m − 1 and dim π1−1 (z) ≥ m − 2 for

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each z ∈ J2 . By (∗m−1 ), we can find a point y2 ∈ J2 such that dim π1−1 (y2 ) = m − 2. If we continue this procedure, we can prove that for any 1 ≤ i < dim P = m there is a point z ∈ G such that dim π1−1 (z) = i. Then Theorem 3.2 follows from Proposition 3.1. 2 In fact, by the proof of Theorem 3.2, we have the following more general result. Theorem 3.3. Let G be a graph and let f : G → 2G be an upper semi-continuous function. Suppose that the inverse limit X = ← lim {G, f } satisfies the condition that dim X < ∞ and if U is any open set of X −− with dim U = i ≥ 1, U can not contain uncountable mutually disjoint i-dimensional subsets. Then X is a dimensionally stepwise space. Corollary 3.4. (Nall [8, Theorem 5.5] and [14, Theorem 3.2]) No inverse limit with a single upper semicontinuous bonding function on a graph can be an n-cell (n ≥ 2). From now, we consider the case that the inverse limit X = ← lim {G, f } is not a polyhedron. We need the −− following lemma. Lemma 3.5. Suppose that G is a graph and f : G → 2G is an upper semi-continuous function such ˜ ({G, f }) < ∞. Then lim{G, f } is finite-dimensional and for any 1 ≤ n < that dim D1 (f −1 ) ≤ 0 and W ←−− dim lim {G, f } there is a point y ∈ G such that dim π1−1 (y) = n. ←−− ˜ ({G, f }) < ∞, by Theorem 2.5 we see that lim{G, f } is finite-dimensional. ˜ Proof. Since I({G, f }) ≤ W ←−− For any natural number m ≥ 2, we will prove the following claim C(m). C(m): If there is a point y in G such that dim π1−1 (y) = m, then there is a point y  ∈ G such that dim π1−1 (y  ) = m − 1. Suppose, on the contrary, that for any x ∈ G, dim π1−1 (x) = m −1. Let y1 ∈ G such that dim π1−1 (y1 ) = m. We choose a Cantor m-dimensional manifold H in π1−1 (y1 ). Let m1 ∈ N such that πm1 +1 (H) is nondegenerate and πi (H) is degenerate for each 1 ≤ i ≤ m1 , i.e., π[1,m1 ] (H) is degenerate and π[1,m1 +1] (H) is non-degenerate. Put π[1,m1 ] (H) = (y1 , y2 , ..., ym1 ). Let J1 be a free arc in Int(πm1 +1 (H)). Then we may assume that −1 dim π[1,m (y1 , y2 , ..., ym1 , x) = dim π1−1 (x) ≥ m − 1 1 +1]

for each x ∈ J1 (see the proof of Proposition 3.1), and hence by the assumption, −1 dim π[1,m (y1 , y2 , ..., ym1 , x) = m. 1 +1]

Let L = {Lj |j ∈ N} be a countable family of arcs in G satisfying that for any nonempty open set V of G, there is Lj ∈ L with Lj ⊂ V . For k, j ∈ N, let J(k, j) be the set of all x ∈ J1 such that there is a −1 Cantor m-dimensional manifold Hx of π[1,m (y1 , y2 , ..., ym1 , x) (∼ = π1−1 (x)), and πi (Hx ) is degenerate for 1 +1] 1 ≤ i < k, πk (Hx ) contains Lj . Note that J1 =

 k,j∈N

J(k, j) =



Cl(J(k, j)).

k,j∈N

By the Baire Category theorem, we can choose k, j ∈ N such that Cl(J(k, j)) contains a nonempty open set, hence dim Cl(J(k, j)) = 1. Put m1 = k and we can choose a point ym1 ∈ Int(Lj ) such that dim π1−1 (ym1 ) ≥  m − 1 (see the proof of Proposition 3.1). By the assumption, dim π1−1 (ym1 ) = m. Then f (m1 −(m1 +1)) (ym1 ) ⊃  J(k, j) and hence f (m1 −(m1 +1)) (ym1 ) ⊃ Cl(J(k, j)). Then we can choose ym2 ∈ G such that m1 ≤ m2 ,

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ym1
 ym1 ← ym2
, and there is a free arc J2 in f −1 (ym2 ) such that dim π1−1 (z) = m for each z ∈ J2 . If we continue this procedure, we obtain a sequence of natural numbers m1 < m1 ≤ m2 < m2 ≤ m2 < · · · , and an infinite weak inverse expand-contract sequence ym1
 ym2
 ym3
 · · ·
 ymk−1
 ymk
· · · ˜ ({G, f }) = ∞. This is a contradiction. Consequently, the claim C(m) is true. Consider in {G, f }. Then W the map π1 : lim {G, f } → G. By the Hurewicz’s theorem (see [5, p. 242]), we can find a point y ∈ G such ←−− that dim π1−1 (y) ≥ dim ← lim {G, f } − 1. By use of the claim C(m), we see that for any 1 ≤ n < dim ← lim {G, f } −− −− there is a point y ∈ G such that dim π1−1 (y) = n. 2 Theorem 3.6. Suppose that G is a graph and f : G → 2G is an upper semi-continuous function such that ˜ ({G, f }) < ∞. Then X = lim{G, f } is a dimensionally stepwise space. dim D1 (f −1 ) ≤ 0 and W ←−− Proof. This theorem follows from Proposition 3.1 and Lemma 3.5. 2 Example 2. (1) Let f : I → C(I) be the surjective upper semi-continuous function defined by f (x) = 0 (x ∈ [0, 1/3)), f (1/3) = [0, 1/3], f (x) = 1/3 (x ∈ (1/3, 2/3)), f (2/3) = [1/3, 2/3], f (x) = 2/3 (x ∈ (2/3, 1)), f (1) = [2/3, 1]. Note that 031/332/3
is a maximal weak inverse expand-contract sequence in {I, f }, and 1/332/3
˜ ˜ is a maximal (inverse) expand-contract sequence in {I, f }. We see that I({I, f }) = 2 = J({I, f }), ˜ ({I, f }) = 3 and {I, f } satisfies the condition of Theorem 3.6. Hence lim{I, f } is a dimensionally stepwise W ←−−

space. In fact, ← lim {I, f } is a 3-cell with a fin (see [8, Example 5.4]). −− (2) Let f : I → C(I) be the surjective upper semi-continuous function defined by f (0) = I and f (x) = 0 (x ∈ (0, 1]). In this case, we have the inverse expand-contract sequence with infinite length as follows: 030303 · · · ˜ ({I, f }) = ∞. We see that lim{I, f } contains a Hilbert cube and it has no finite dimensional Note that W ←−− non-degenerate open sets and hence it is not a dimensionally stepwise space (see [8, Example 2.3]). 4. Inverse limits with set-valued functions and ANRs In this section, we assume some familiarity with shape theory (see [2] and [4] for shape theory). Let X be a continuum contained in a metric space M . Then X is weak homotopically trivial within small neighborhoods of M provided that if f : S n → X is any map from the n-sphere S n (n ≥ 0) to X, then f is null-homotopic in any neighborhood of X in M . Note that if X is an FAR (i.e., X has trivial shape. See [2,4]), then X is weak homotopically trivial within small neighborhoods of any ANR M .

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We consider the following property (*) of X; there exists a sequence {Vn }n=0,1,2,... of finite closed coverings of X such that (i) V0 = {X}, and X = ∪{IntX V | V ∈ Vn } for each n, (ii) limn→∞ mesh(Vn ) = 0, (iii) if Vα ∈ V = ∪n Vn and ∩α Vα = ∅, then ∩α Vα is weak homotopically trivial within small neighborhoods of M (see [12]). Also we consider the following property local (*) of X; there exists a sequence {Vn }n=1,2,... of finite closed coverings of X such that (i) X = ∪{IntX V | V ∈ Vn } for each n, (ii) limn→∞ mesh(Vn ) = 0, (iii) if Vα ∈ V = ∪n Vn and ∩α Vα = ∅, then ∩α Vα is weak homotopically trivial within small neighborhoods of M (see [12]). We need the following propositions. Proposition 4.1. (see [12, (3.2) Lemma]) Suppose that X is a continuum contained in a metric space M . If X has the property local (*), then X is an ANR (=absolute neighborhood retract). Moreover, if X has the property (*), then X is an AR (=absolute retract). Proposition 4.2. Let Xi (i ∈ N) be a sequence of finite-dimensional compacta and let fi,i+1 : Xi+1 → 2Xi −1 be a surjective upper semi-continuous function for each i ∈ N such that fi,i+1 is cell-like. Then the inverse limit ← lim {Xi , fi,i+1 } is shape equivalent to X1 . Moreover if X1 is an FAR, then ← lim {Xi , fi,i+1 } is also an −− −− FAR. Proof. Consider the inverse sequence X1 ← G(f ; 1, 2) ← G(f ; 1, 2, 3) ← · · · whose bonding maps pn,n+1 : G(f ; 1, 2, ..., n + 1) → G(f ; 1, 2, ..., n) are natural projections defined by pn,n+1 (x1 , x2 , ..., xn , xn+1 ) = (x1 , x2 , ..., xn ). Since the projections p−1 n,n+1 are cell-like, pn,n+1 : G(f ; 1, 2, ..., n + 1) → G(f ; 1, 2, ..., n) induces a shape equivalence. Hence we see that the inverse limit lim {G(f, 1, ..., i), pi,i+1 } = ← lim {Xi , fi,i+1 } is shape equivalent to X1 . If X1 is an FAR, then ← lim {Xi , fi,i+1 } ←−− −− −− is also an FAR. 2 In [8], Ingram gave many examples of inverse sequences of the unit interval I with upper semi-continuous set-valued functions whose inverse limits are dendrites. We need the following condition. Let f : X → 2Y be an upper semi-continuous function. Consider the condition Z(f ) for f . Z(f ): For any x ∈ X, y ∈ Y with y ∈ f (x), any closed neighborhood A of x in X and any closed neighborhood B  of y in Y , there are a closed connected neighborhood A of x in X and a closed connected neighborhood B of y in Y such that A ⊂ A , B ⊂ B  and the pair (B, A) satisfies the condition c(B, A); for any continuum K (⊂ A) with x ∈ K, the set C(B, A; K) = {z ∈ B | f −1 (z) ∩ K = ∅} (= f (K) ∩ B) is connected. The main theorem of this section is the following. Theorem 4.3. Let Gi (i ∈ N) be a sequence of graphs and let fi,i+1 : Gi+1 → 2Gi be a surjective up−1 −1 per semi-continuous function for each i ∈ N such that fi,i+1 is cell-like (i.e., fi,i+1 (y) is a tree for each Gi y ∈ Gi ). Suppose that each fi,i+1 : Gi+1 → 2 satisfies the condition Z(fi,i+1 ). Then the inverse limit lim {Gi , fi,i+1 } of the inverse sequence {Gi , fi,i+1 } is an ANR which is homotopic to G1 . Moreover, if G1 ←−− −1 ˜ is a tree, then ← lim {Xi , fi,i+1 } is an AR. Especially, if dim D1 (fi,i+1 ) ≤ 0 (i ∈ N) and I({G i , fi,i+1 }) = 0, −− then ← lim {Gi , fi,i+1 } is a dendrite (=1-dimensional AR). −−

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Proof. In the proof, we use the fact that the intersection of continua (=trees) contained in a tree is an empty set or a tree. Suppose that  > 0 is a very small positive number. Let n ∈ N and (x1 , x2 , ..., xn ) ∈ G(f ; 1, 2, ..., n). −1 −1 Since fi,i+1 (xi ) (i = 1, 2, ..., n − 1) is a tree in Gi+1 , we choose a closed neighborhood Ti+1 of fi,i+1 (xi ) in Gi+1 such that Ti+1 is a tree. Also, we choose a closed neighborhood Bi (i = 1, 2, ...n) of xi in Gi such −1 that Bi is a tree such that Bi+1 ⊂ Ti+1 (i = 1, 2, ..., n − 1), diam Bi ≤  and fi,i+1 (Bi ) ⊂ Ti+1 for each i = 1, 2, ..., n − 1. Put V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; ) = {(zi )∞ {Xi , fi,i+1 } | zi ∈ Bi (i = 1, 2, ..., n)}. i=1 ∈ lim ←−− Moreover, by use of the property Z(fi,i+1 ), we can choose closed neighborhoods Bn , Bn−1 , ..., B1 such that V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; ) is an FAR. First, we choose a small closed connected neighborhood Bn of  xn in Gn which is a tree and a small closed connected neighborhood Bn−1 of xn−1 in Gn−1 such that the pair    (Bn−1 , Bn ) satisfies the condition c(Bn−1 , Bn ). Inductively, we have pairs (Bi−1 , Bi ) (i = n − 1, n − 2, ...., 2)  such that Bi and Bi are small closed connected neighborhoods of xi in Gi , Bi ⊂ Bi (i = n − 1, n − 2, ..., 2)   , Bi ) satisfies the condition c(Bi−1 , Bi ). Put B1 = B1 . Let Cn = Bn and let Cn−1 = and the pair (Bi−1   C(Bn−1 , Bn ; Cn ) ∩ Bn−1 , Cn−2 = C(Bn−2 , Bn−1 ; Cn−1 ) ∩ Bn−2 . If we continue this procedure inductively, we have the sequence Ci (i = n, n − 1, n − 2, ..., 1) of trees such that xi ∈ Ci . We will show that V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; ) = ← lim {Yi , gi,i+1 }, −− −1 −1 −1 −1 where Y1 = C1 , Y2 = f1,2 (Y1 ) ∩ C2 , Y3 = f2,3 (Y2 ) ∩ C3 , · · · , Yn = fn−1,n (Yn−1 ) ∩ Cn , Yi = fn,i (Yn ) (i ≥ n) Yi and gi,i+1 : Yi+1 → 2 is the set-valued function defined by gi,i+1 (z) = Yi ∩ fi,i+1 (z) for z ∈ Yi+1 . By the definitions, we see that V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; ) ⊃ ← lim {Yi , gi,i+1 }. We will show the converse −− inclusion. Let

y = (yi ) ∈ V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; ).  Since yn ∈ Bn = Cn , then yn−1 ∈ C(Bn−1 , Bn ; Cn ) ∩ Bn−1 = Cn−1 . Since yn−1 ∈ Cn−1 , then yn−2 ∈  C(Bn−2 , Bn−1 ; Cn−1 ) ∩ Bn−2 = Cn−2 . If we continue this procedure, we see that yi ∈ Ci and hence yi ∈ Yi for i ∈ N. This implies that y ∈ lim {Yi , gi,i+1 }. Hence ←−−

V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; ) = ← lim {Yi , gi,i+1 }. −− −1 −1 −1 Note that for x ∈ Yi (i = 1, 2, ..., n −1), gi,i+1 (x) = fi,i+1 (x) ∩Yi+1 (⊂ Ti+1 ). Hence gi,i+1 is cell-like for i ∈ N. Since Y1 = C1 is a tree, by Proposition 4.2 ← lim {Yi , gi,i+1 } is an FAR. Hence V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; ) −− is an FAR. Let 1 > 2 > 3 > · · · be a sequence of positive numbers with limi→∞ i = 0. For n ∈ N, there is a finite set Fn of G(g : 1, 2, ..., n) such that

lim {Xi , fi,i+1 } = ∪{V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; n ) | (x1 , x2 , ..., xn ) ∈ Fn }. ←−− Put Vn = {V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; n ) | (x1 , x2 , ..., xn ) ∈ Fn }. By the definitions of V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; n ), we see that the sequence {Vn }n=1,2,... is a family of finite closed coverings of ← lim {Xi , fi,i+1 } satisfying the conditions (i) and (ii) of local (*). Note that −− if V = V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; n ) ∈ {Vn }n=1,2,... , then V can be represented by the inverse limit lim {Yi , gi,i+1 } as above. If n ≤ n and ←−−

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V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; n ), V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn  ; n ) ∈ {Vn }n=1,2,... , then we see that V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn ; n ) ∩ V (x1 , x2 , ..., xn ; B1 , B2 , ..., Bn  ; n ) = {(zi )∞ {Xi , fi,i+1 } | zi ∈ Bi ∩ Bi (i = 1, 2, ..., n) and zj ∈ Bj (j = n + 1, ..., n )} i=1 ∈ lim ←−− is an empty set or an FAR, because that it can be represented by an inverse limit ← lim {Zi , gi,i+1 }, where −− −1 Z1 is a tree and gi,i+1 is cell-like. Note that the intersection of decreasing sequence of FARs is also an FAR. By using these arguments, moreover we see that {Vn }n=1,2,... also satisfies the condition (iii) of local (*). By Proposition 4.1, ← lim {Gi , fi,i+1 } is an ANR. By Proposition 4.2, we see that the inverse limit −− lim {G , f } is shape equivalent to G1 and hence it is homotopy equivalent to G1 . Moreover, if G1 is i i,i+1 ←−− −1 a tree, then ← lim {Gi , fi,i+1 } is a contractible ANR and hence an AR. If dim D1 (fi,i+1 ) ≤ 0 (i ∈ N) and −− ˜ I({G , f }) = 0, then lim {G , f } is 1-dimensional and hence it is a dendrite (=1-dimensional AR). i i,i+1 i i,i+1 ←−− This completes the proof. 2 Let K be any finite simplicial complex in I × I and let f : I → 2I be the upper semi-continuous function defined by G(f ) = |K|. Then f satisfies the condition Z(f ). Hence we have the following result. Theorem 4.4. Suppose that Ii (i ∈ N) is a sequence of the unit interval I = [0, 1] and Ki is a finite simplicial complex in Ii × Ii+1 satisfying that for any x ∈ Ii+1 , (Ii × {x}) ∩ |Ki | = ∅ and for any y ∈ Ii , ({y} × Ii+1 ) ∩ |Ki | is a nonempty connected set (=a closed interval). Let fi,i+1 : Ii+1 → 2Ii be the surjective upper semi-continuous function defined by G(fi,i+1 ) = |Ki |. Then ← lim {I , f } is an AR. Moreover, if −− i i,i+1 ˜ i , fi,i+1 }) = 0, then lim{Ii , fi,i+1 } is a dendrite. dim |Ki | ≤ 1 (i ∈ N) and I({I ←−− Corollary 4.5. If f : G → 2G is a surjective upper semi-continuous function on a graph G such that ˜ ({G, f }) < ∞, f −1 is cell-like and f satisfies Z(f ), then the inverse limit lim{G, f } with dim D1 (f −1 ) ≤ 0, W ←−− the single upper semi-continuous bonding function f is a dimensionally stepwise ANR which is homotopic to G. Example 3. (1) In the statement of Theorem 4.3, we need the condition Z(fi,i+1 ). Let g : I = [0, 1] → I be the map defined by g(x) =

π x (1 + sin ) 2 2x

for x ∈ (0, 1] and g(0) = 0. Let f = g −1 : I → 2I and h : I → 2I be the surjective upper semi-continuous function defined by h(x) = 0 (x ∈ [0, 1)) and h(1) = I. Consider the inverse sequence {Ii , fi,i+1 } defined by −1 f1,2 = f, f2,3 = h, fi,i+1 = id (i ≥ 3). Note that fi,i+1 is cell-like, each graph G(fi,i+1 ) is homeomorphic to an arc, and hence locally connected. But it does not satisfies the condition Z(f1,2 ). For the points x = 0, y = 0, the set C(B, A; K(= {0})) = {z ∈ B | f −1 (z) ∩ {0} = ∅} is not connected for any neighborhood A of x = 0 lim {I , f } is homeomorphic to the following set and any neighborhood B of y = 0. In fact, we see that ← −− i i,i+1 3 X in the Euclidean 3-space R : X = {(x, y) ∈ R2 |x ∈ I, y = g(x)} ∪ S × [0, 1] (⊂ R2 × R) where S = {(x, 0) |x ∈ I, g(x) = 0}. Note that ← lim {I , f } is not locally connected and hence not an −− i i,i+1 ANR.

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(2) Let f : I = [0, 1] → 2I be the upper semi-continuous function defined by f (x) = {0, 1} (x ∈ I). Note that f is not surjective, f satisfies the condition Z(f ) and f −1 (0), f −1 (1) are arcs. But ← lim {I, f } is a Cantor −− set and hence not an ANR (see [8, Example 1.2]). (3) Let f : I → 2I be the surjective upper semi-continuous function defined by f (x) = {0, 1} (x = 1/2) and ˜ f (1/2) = I. Note that f satisfies the condition Z(f ), f −1 is cell-like, dim D1 (f −1 ) ≤ 0 and I({I, f }) = 0. Hence ← lim {I, f } is a dendrite. In fact, lim {I, f } is a dendrite with a Cantor set of endpoints (see [8, Example −− ←−− 2.22]). (4) Let n ∈ N with n ≥ 2 and let f : I → C(I) be the surjective upper semi-continuous function defined by f (x) = 0 (x ∈ [0, 1/n)) and for 1 ≤ i ≤ n − 1, f (i/n) = [(i − 1)/n, i/n], f (x) = i/n (x ∈ (i/n, (i + 1)/n)), f (1) = [(n − 1)/n, 1]. Then 031/n32/n3 · · · 3(n − 1)/n
˜ ˜ is a maximal weak inverse expand-contract sequence in {I, f }. Note that I({I, f }) = J({I, f }) = n − 1, −1 ˜ is cell-like and f satisfies the condition Z(f ). We see that ← lim {I, f } is n-dimensional W ({I, f }) = n, f −− and a dimensionally stepwise AR. In fact, the space is a polyhedron (cf. [8, Example 5.4]). Finally, we have the following problems (see also [8, Chapter 6, Problems]). Problem 4.6. Let f : G → 2G be an upper semi-continuous function on a graph G such that dim lim {G, f } < ∞. Is the inverse limit ← lim {G, f } a dimensionally stepwise space? ←−− −− Problem 4.7. Is it true that the Menger curve can be obtained as an inverse limit with a sequence of upper semi-continuous bonding functions on the interval [0, 1]? References [1] I. Banič, On dimension of inverse limits with upper semicontinuous set-valued bonding functions, Topol. Appl. 154 (2007) 2771–2778. [2] K. Borsuk, Theory of Shape, Monografic Matematyczne, vol. 59, Polish Scientific Publishers, Warszawa, 1975. [3] W. Charatonik, R. Roe, Inverse limits of continua having trivial shape, Houst. J. Math. 38 (2012) 1307–1312. [4] J. Dydak, The Whitehead and Smale theorems in shape theory, Diss. Math. 156 (1979). [5] R. Engelking, Theory of Dimensions Finite and Infinite, Heldermann Verlag, Lemgo, 1995. [6] S. Greenwood, J. Kennedy, Connectedness and Ingram–Mahavier products, Topol. Appl. 166 (2014) 1–9. [7] A. Illanes, A circle is not the generalized inverse limit of a subset of [0, 1]2 , Proc. Am. Math. Soc. 139 (2011) 2987–2993. [8] W.T. Ingram, An Introduction to Inverse Limits with Set-Valued Functions, Springer Briefs in Mathematics, Springer, New York, 2012. [9] W.T. Ingram, Concerning dimension and tree-likeness of inverse limits with set-valued functions, Houst. J. Math. 40 (2014) 621–631. [10] W.T. Ingram, W.S. Mahavier, Inverse Limits: From Continua to Chaos, Developments in Mathematics, vol. 25, Springer, New York, 2012. [11] Hisao Kato, On dimension and shape of inverse limits with set-valued functions, Fundam. Math. 236 (2017) 83–99. [12] Hisao Kato, Limitting subcontinua and Whitney maps of tree-like continua, Compos. Math. 66 (1988) 5–14. [13] W.S. Mahavier, Inverse limits with subsets of [0, 1] × [0, 1], Topol. Appl. 141 (2004) 225–231. [14] V. Nall, Inverse limits with set valued functions, Houst. J. Math. 37 (2011) 1323–1332. [15] V. Nall, The only finite graph that is an inverse limit with a set valued function on [0, 1] is an arc, Topol. Appl. 159 (2012) 733–736.