IMFM
Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia
Preprint series Vol. 51 (2013), 1186
ISSN 2232-2094
INDUCING FUNCTIONS BETWEEN INVERSE LIMITS WITH UPPER SEMICONTINUOUS BONDING FUNCTIONS
Iztok Banic Matevž Crepnjak Goran Erceg Matej Merhar Uros Milutinovic
Ljubljana, May 17, 2013
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Inducing functions between inverse limits with upper semicontinuous bonding functions
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Iztok Banič, Matevž Črepnjak, Goran Erceg, Matej Merhar and Uroš Milutinovic
University of Maribor, Slovenia University of Split, Croatia
Abstract
In this paper we introduce the category CU in which the compact metric spaces are objects and upper semicontinuous functions from X to 2Y are morphisms from X to Y. We also introduce the category ICU of inverse sequences in CU. Then we investigate the induced functions between inverse limits of compact metric spaces with upper semicontinuous bonding functions. We provide criteria for their existence and prove that under suitable assumptions they have surjective graphs. We also show that taking such inverse limits is very close to being a functor (but is not a functor) from ICU to CU, if morphisms are mapped to induced functions. At the end of the paper we give a useful application of the mentioned results.
Keywords: Inverse limits, Upper semi-continuous functions, Induced functions, Induced morphisms
2000 Mathematics Subject Classification: 54F15,54C60
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1 Introduction
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A function f : X ^ 2Y, where X and Y are compact metric spaces, is upper semicontinuous function from X to 2Y (abbreviated u.s.c. function) if for each open set V C Y the set {x G X | f (x) C V} is an open set in X. We will interpret such a function f : X ^ 2Y as a morphism f : X ^ Y and thus obtain a category, which we will denote by CU. Then ICU is the standard category of inverse sequences in CU.
Consider two inverse sequences	and	of compact
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. - njn=1	L n i/n Jn= 1
metric spaces and morphisms of CU. Let for each positive integer i, be a u.s.c. function from X, to 2Yi. In this paper we study the question which u.s.c. functions from Hin{Xn,/n}^c=1 to 21—{Yn'gn}n=1 can be interpreted as induced by <£1; <^2, <^3, ...; we also study the problem of existence of such induced functions as well as their properties. The special case when <^3, ... are single-valued continuous functions was studied in [8, 15]. In their paper [15], Ingram and Mahavier showed that if X and Y are inverse limits of inverse sequences with u.s.c. bonding functions and each : X, ^ Y, is a homeomorphism, then the function induced by the functions is a home-omorphism. In [8] it was shown under suitable assumptions, if each of the functions are surjective, one-to-one, or a homeomorphism respectively, then also the induced mapping is surjective, one-to-one, or a homeomor-phism, respectively.
In the present paper we assume that each ^ is a u.s.c. function from Xj to 2Yi and show that even in such more general situation the notion of an induced function
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$ : l^m{Xn, /n}~=1 ^ 2in{Yn'fln}n=1
can be defined in such a way that it is a u.s.c. function (if certain mild conditions are satisfied). Further we show that if each of the ^,'s has a surjective graph, then under certain additional conditions also $ has a surjective graph.
It is a well-known fact that if {Xn,/n}^=1 and {Yn,gn}^=1 are inverse sequences of compact metric spaces and continuous single-valued bonding functions, and if for each positive integer i, is a continuous single-valued function from X,- to Yj, then the transformation
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{Xn, /n}n=
1 I-> ^lin{Xn, fn} n=1
and
-—> In-
is a functor from the category of inverse sequences in the category of compact metric spaces with continuous single-valued functions (i.e. from the category in which inverse sequences of compact metric spaces with continuous single-valued bonding functions are objects, and sequences (^1, <^2, <^3,...) of single-valued mappings having certain commutativity property are morphisms), to the category of compact metric spaces and continuous functions.
In the present paper we prove that in the case when the inverse sequences {Xn,/n}^=1 and {Yn,gn}^=1 are inverse sequences in CU (i.e. Xn, Yn are CD	compact metric spaces, and /n : Xn+1 ^ Xn, gn : Yn+1 ^ Yn are morphisms
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in CU, meaning that fn : Xn+1 ^ 2Xn, gn : Yn+\ ^ 2Yn are u.s.c. functions) and each ^ is a u.s.c. function from Xj to 2Yi, the transformation
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and
Ctf

{Xn, fn}n= 1 I-> l^im{Xn, fn} n=1
(^1, ...) -—> lim^
is not a functor from the category ICU of inverse sequences in CU to the category CU, but is very close to being one.
In the last section we give a useful application of the mentioned results.
2 Definitions and notation
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Our definitions and notation mostly follow Nadler [19], Ingram [14], and Ingram and Mahavier [15].
A map is a continuous function. A continuum is a nonempty, compact and connected metric space. 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 E 2X
(N
Nd(e, A) = {x E X | d(x, a) < e for some a E A}.
The set 2X will be always equipped with the Hausdorff metric Hd, which is defined by
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Hd(H,K) = inf{e > 0 | H C Nd(e,K),K C Nd(e, H)},
for H, K E 2X. Then (2X, Hd) is a metric space, called the hyperspace of the space (X,d). For more details see [13, 19].
For a function f : X ^ 2Y and a subset A C X, we use f [A] = (Jx&A f (x) to denote the image of A under f.
The graph r(f) of a function f : X ^ 2Y is the set of all points (x,y) E X x Y such that y E f (x) (as defined in [14, p. 2]).
A function f : X ^ 2Y has a surjective graph if for each y E Y there is an x E X, such that y E f (x), i.e. if f [X] = Y (as defined in [14, p. 2]).
A function f : X ^ 2Y, where X and Y are compact metric spaces, is upper semicontinuous function from X to 2Y (abbreviated u.s.c.) if for each open set V C Y the set {x E X | f (x) C V} is an open set in X.
The following is a well-known characterization of u.s.c. functions (see [15, p. 120, Theorem 2.1]).
Theorem 2.1. Let X and Y be compact metric spaces and f : X ^ 2Y a function. Then f is u.s.c. if and only if its graph r(f) is closed in X x Y.
A sequence {Xk, fkis an inverse sequence of compact metric spaces and u.s.c. bonding functions if each Xk is a compact metric space and each fk is a u.s.c. function fk : Xk+1 ^ 2Xk.
The inverse limit of an inverse sequence {Xk, fk}£=1 of compact metric spaces and u.s.c. bonding functions is defined to be the subspace of the product space	Xk of all x = (x1,x2,x3,...) G nXk, such that
xk G fk(xfc+1) for each k. The inverse limit of {Xk, fk}£=1 is denoted by
Jm{Xfc ,fk }r=1.
In the present paper we will interpret inverse sequences {Xk, fk}£=1 of compact metric spaces and u.s.c. bonding functions as inverse sequences in CU and study hm as a possible functor from ZCU to CU.
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 [15]. 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, 7, 9, 12,
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16, 17, 11, 18, 20, 21, 22, 23, 24], as well as the book [14].
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
t^./ \	fdn(xn,yn)
D(x,y)= sup '
n
n€{1,2,3,...} I 2
CO	where x = (x1,x2, x3,...), y = (y1 ,y2,y3,...). It is well known that the
<x
metric D induces the product topology on JJ^Xn [10, p. 190]. The set N denotes the set of all positive integers {1, 2, 3,...}.
3 The categories CU and ICU
In this section we give detailed descriptions of the following two categories:
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1.	CU: the category of compact metric spaces and u.s.c. functions;
2.	ICU: the category of inverse sequences in CU.
For any category K, the class of objects of K is denoted by Ob(K), and for a	any X, Y G Ob(K), the set of morphisms of K from X to Y is denoted by
Mor(K)(X,Y).
3.1 CU
CM
The category CU of compact metric spaces and u.s.c. functions consists of the following objects and morphisms:
1. Ob(CU) is the class of compact metric spaces;
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s
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CO CD
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CD CO
CD
cu
2. The set of morphisms Mor(CU)(X, Y) from X to Y is the set of u.s.c. functions X ^ 2Y. The u.s.c. function f : X ^ 2Y as a morphism of
CU is denoted by f : X ^ Y. This can be reformulated as
Mor(CU)(X, Y) = {f : X ^ Y | f : X ^ 2Y is u.s.c.}.
We also define the partial binary operation o (the composition) on the class of morphisms of CU as follows. For each f G Mor(CU)(X, Y) and each g G Mor(CU)(Y, Z) we define g o f G Mor(CU)(X, Z) by
o	(g o f)(x) = g[f (x)]= U g(y)
ye/(x)
for each x G X. It is easy to see that for each u.s.c. function f : X ^ 2Y and each u.s.c. function g : Y ^ 2Z, the composition g o f defined above is again a u.s.c. function X ^ 2Z [14, p. 4]. That means that if f : X ^ Y and g : Y ^ Z, then g o f : X ^ Z. Therefore o is well-defined. It is also easy to see that o is associative.
Theorem 3.1. CU is a category.
Proof. All that is left to show is that for each X G Ob(CU) there is a morphism 1X : X ^ X such that 1X o f = f and g o 1X = g for any morphisms f : Y ^ X and g : X ^ Z. We easily see that the u.s.c. function 1X : X ^ 2X, defined by 1X(x) = {x} for each x G X satisfies the above conditions.	□
3.2 ICU
The sequence {Xn,fra}^=1 is an inverse sequence in CU if each Xn is an object of CU and each fn is a morphism from Mor(CU)(Xn+1 , Xn) (i.e. fn : Xn+1 ^ Xn, meaning that fn is a u.s.c. function fn : Xn+1 ^ 2Xn).
The category ICU of inverse sequences in CU consists of the following objects and morphisms:
1.	Ob(ZCU) is the class of inverse sequences {Xn,	in CU;
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2.	For any two objects X = {Xn,/n}~=1 and Y = {Yn,gn}~=1 of ICU, the set Mor(ZCU)(X, Y) consists of all sequences v =	• • •), where each ^ is a morphism in CU from Xj to Y (i.e. it is a u.s.c. function from Xj to 2Yi), such that gj o = ^ o /j for each positive integer i.
We also define the partial binary operation □ (the composition) as follows. Let X = {Xn,/n}~=i, Y = {Yn,gn}~=i, Z = {Zn,hn}~=i be any objects of ICU, and let v = (^1, V2, V3, • • •) G Mor(ICU)(X,Y), and ^ = • • •) G Mor(ZCU)(Y, Z) be any morphisms of ICU. Then we define by
= (^1 o ^1,^2 o ^2,^3 o ^3, • • •)•


1
csr CO csr csr
u
a
CD U
Theorem 3.2. ICU is a category.
Proof. Using the above notation we see that for each i, o : Xj ^ 2Zi is a u.s.c. function, i.e. o : Xj ^ Zj is a morphism of CU. Next we see that for each positive integer i,
hj o (^j+1 o <£j+1) = (^j o ^j) o /j
since
hj o (^j+1 o ^j+1) = (hj o ^j+1) o <£j+1 = (^j o gj) o ^j+1 =
^j o (gj o ^j+1) = ^j o (^j o /j) = (^j o ^j) o /j^
CO	That proves that G Mor(ICU)(X,Z), as required. Since o is associa-
tive, it easily follows that □ is also associative.
Also, from the properties of functions 1Xi it easily follows that for any inverse sequence X = {Xn, /n}n=1 the morphism 1x = (1Xl, 1X2, 1X3, • • •) : X ^ X satisfies the conditions
HH
1x□v = V and	=
m
for any morphisms v : {Y^, gn}n=1 ^ X and : X ^ {Zn, hn}n=1.	□
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4 Induced functions and induced morphisms
We begin this section with the definition of functions induced by sequences of u.s.c. functions.
Definition 4.1. Let {X^/^n^ and {in,gra}n=i be any inverse sequences of compact metric spaces and u.s.c. bonding functions (i.e. inverse sequences
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in CU), and let for each n, : Xn ^ 2Yn be a u.s.c. function. Let for each (Xi,X2,X3, . . .) G l^im{Xn, /n}n=1,
$(Xi,X2,X3,...) = (^i(xi) X ^2(^2) X ^3(^3) X ••• ) n lim{Yn,gn}n=1- (1)
limiY g
If $ is a u.s.c. function from lim{Xn, /n}n=i to 2l nJn=1, then we say that it is induced by	, ...).
vD	Theorem 4.2 provides a simple criterion for recognizing induced functions.
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Theorem 4.2. We use the notation from Definition 4.1. Then $ is induced by <^2, <^3,...) if and only if $(x^ x2, x3,...) = 0 for each (x^ x2, x3,...) G Jm{Xn,/n}~i. Proof. Let us denote
h = iim{X/}i=i
and
K = !m{Y,g}=i.
If $ : H ^ 2K is induced by <^2, <^3,...), then for each (x^ x2, x3,...) G H, $(xi, x2, x3,...) G 2K. Therefore $(x^ x2, x3,...) = 0.
Now assume that $(xi, x2, x3,...) = 0 for each (x^ x2, x3,...) G H. Since $(xi,x2,x3,...) is compact it is closed in K. Therefore the function $ : H ^ 2k is well-defined.
Next we prove that $ is a u.s.c. function. It is sufficient to prove that the graph r ($) of $ is a closed subset of H X K. It follows from the definition S	of $ that
r($) = {((xi,x2,x3,...), (yi,y2,y3,...)) g H x K | Vi g N,y g ^ (x)}. Let a : H X K ^ (Xi X Yi) X (X2 X Y2) x ■ ■ ■ be defined by
a (x,y) = ((xi,yi), (x2,y2),...),
for all x = (x^ x2, x3,...) G H and y = (yi; y2, y3,...) G K.
Note that A : H X K ^ Im a defined by A (x, y) = a (x, y) is a homeo-morphism.
We prove that A (r ($)) is closed in Im A = A(H X K).
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A (r($)) = {((xi,yi), (x2,y2),...) | x G H,y G K, Vi G N,y G ^i (xi)}
= {((xi, yi), (x2,y2),...) | x G H,y G K, Vi G N, (x^) G r(^i)} = {((xi, yi), (x2, y2),...) G r (vi) x r (^2) x ■ ■ ■ | x G H, y G K} = (r(vi) X r(v2) X •••) n{((xi,yi), (x2,y2),...) | x G H,y G K}
= (r(vi) X r(v2) X---) n a(h x k).
The product
rM x r(^2) x r(^s) x---
is a closed subset of (X1 x Yi) x (X2 x Y2) x ■ ■ ■ , therefore A (r ($)) is closed in A(H x K). It follows that r ($) is closed in H x K.	□
The next theorem presents a commutativity-like condition under which $ is induced.
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Theorem 4.3. Let {Xn, /ra}n=1 and |Yn,gra}nL1 be any inverse sequences of compact metric spaces and u.s.c. bonding functions (i.e. inverse sequences in CU), and let for each n, : Xn ^ 2Yn be a u.s.c. function. If
<n[/n(x)] C g„[^„+1(x)]
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for each positive integer n and each x G Xn+1, then $ defined by (1) is induced by (<1; <2, <3,...).
Proof. By Theorem 4.2 it suffices to prove that $(x) is nonempty for arbitrary x G Jm{Xra, /„}n=1.
For arbitrary x G ^m{Xra,	we construct a point y = (y1, y2, y3,...) G
$(x) by an inductive construction of coordinates y^ More precisely, by induction on i G N, we construct a sequence yi G Y satisfying yi G < (x») and yi G g (yi+1) for each i.
We choose any y1 G ^1(x1); it can be done since ^1(x1) is nonempty. Assume next that we have already constructed yi G < (x»). Now we construct yi+1 G <i+1 (xi+1) such that yi G gi (yi+1).
It follows from x G lim{Xra, /n}n=1 that x» G /(xj+1). Therefore
l-H	1
yi G ^i(xi) C < [/j (xi+1)] C g» [^j+1 (xi+1)] = y gi(t).
Hence, there exists a point t0 G ^i+1 (xi+1) such that yi G gi (to). We take any such t0 for yi+1.	□
This immediately leads to the following corollary.
• 1
Corollary 4.4. Let {Xn,/n}n=1 and {Yn,gn}n=1 be any objects of ICU, and let the sequence (<1; <2, <3,...) be any morphism of ICU from {Xn, /ra}n=1 to {Y„,g„}n=1. Then $ : Jm{Xn,/n}~1 ^ 2l{yn'rf=1, defined by (1), is induced by (<£1, <£2, <£3, • • •), meaning that $ : 1m{Xn, /}n=1 ^ 1im{yra, gra}g=1 is a morphism in CU.	□
Definition 4.5. The function $ from Corollary 4-4 is called the morphism of CU induced by the morphism (^i, <2, <3, • • •) of ZCU and is denoted by $ = ljn <.
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Note that if <2, <3, • • •) is not a morphism of ZCU but the induced function $ is a morphism of CU (that happens when <2, <3, • • •) satisfies the conditions of Theorem 4.2) Jim < does not exist.
The induced morphism Jim < : ¿m{Xra, /«}£= ^ ¿-{Yn, gn}^^ cannot be defined simply by the formula
lim^¿(Xi,X2,X3, • • •) = <i(xi) x <2^2) x <3^3) x
since the right hand side product of (2) is not necessarily a subset of ¿-{Y^, gra}n=i, as shown by Example 4.6.
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Example 4.6. Let X, = Y = [0,1], let / = g, = 1[0 i], where 1[0 i] : [0,1] ^ 2M
is the u.s.c. function, defined by 1[0;i](x) = {x} for each x G [0,1], and let < : [0,1] ^ 2[0,i] be defined by its graph: r (<,) = [0,1] x [0,1], for each positive integers i. Then ^i(xi) x <2(x2) x <3(x3) x ■ ■ ■ is not a subset of
ijm{Y„,g„}n=i-
Proof. Obviously, g, o <i+i = < o / holds true for any positive integer i, and therefore (^i, <2, <3, • • •) is a morphism of ZCU.
Also, ljm{X„, /n}n=i = lim{Yn,gn}n=i = {(*,*,*,•••) I t G [0,1]}, and therefore
^(xi,x2,x3,^ • •) = ^i(xi) x <2^2) x <3(x3) x ■ ■ ■ = [0, 1] x [0, 1] x [0, 1] x ■ ■ ■
l-H
is not a subset of ¿-{Yn,gn}n=i (and therefore it is not an element of
This example shows also that (2) cannot replace (1) in the definition of induced functions.
In the following theorem we prove that if each of the has a surjective graph, then also Jim < has a surjective graph. Note that it is not required that any of the functions /n and gn has a surjective graph.
Theorem 4.7. Let {Xra,/ra}n=i and {Yn,gra}n=i be any objects of ZCU and let the sequence (<i, <2, <£3, • • •) be a morphism of ZCU from {Xn,/n}n=i to {Yn,gra}n=i such that < : X, ^ 2Yi has a surjective graph for each positive integer i. Then lim < has a surjective graph.
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cm
r
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Proof. Let y = (y^y2,y3, • • •) £ ^—{Yl,	be arbitrary. We construct a
point x £ ^im{Xra, /„}„= such that y £ ^im ^(x).
Let n be any positive integer. Since : X2 ^ 2Yn has a surjective graph, there is a point x„ £ X2 such that y2 £ ^2(xl). We choose and fix such an x„. Then by downwards induction we prove that for any k £ {1, 2, 3,...,n — 1} there is x„ £ Xfc such that y^ £ ^ (x„) and x„ £ / (x„+1).
Let k be any integer from {1, 2, 3,...,n — 1}. Assume that x„+1 has already been chosen in such a way that yk+1 £ ^fc+1(x„+1). Note that this assumption is fulfilled for k = n — 1.
Since yfc £ gfc (yfc+1) and yfc+1 £ ^fc+1 (xl+J, it follows that
yfc £ gfc[^fc+1(x„+1)] = ^fc[ffc(x„+1)].
Therefore there is a point x„ £ Xk such that x„ £ /k (x„+1) and yk £ ^fc(x„) and we fix one such x„.
This construction yields
& .........
x — (x1 , x2 , x3 , . . . , xn-1
where £ Xj is arbitrarily chosen for each i > n. Then {x2is a sequence in the compact metric space j=1 Xj, D). Let x = (x1, x2, x3,...) £ nj=1 Xj be any accumulation point of the sequence {x2}2=1.
Next we prove that x £ ^m{Xra,/„}„= and that y £ l—^(x). Let {U2=1 be a strictly increasing sequence of integers such that
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x (xl , x2 , x3 , . . . , xn-1, xn, Zn+1, Zn+2, Zn+3, - - 0 £ J^ J^ Xi
j=1
lim xjn = x.
First we prove that x £ l^m{X2, /2}3=1. Let m be any positive integer. Then (xj+1,xmm) £ r(/m) for each positive integer ik > m. Since ^lim (x^^x^) =
(xm+1, xm) and since the graph r(/m) is closed in Xm+1 x Xm, it follows that (xm+1,xm) £ r(/m). Therefore x £ lim{Xn, /ra}2=1.
Finally we prove that y £ lim ^j(x). Let m be any positive integer. Then
ym £	) for each positive integer ik > m. Therefore (xj,ym) £
for each ik > m. Since lim (xj,ym) = (xm,ym) and since the graph r(^m)
is closed in Xm x Yj, it follows that (xm,ym) £	and therefore ym £
It follows that (y1, y2, y3,...) £ llm ^j(x1, x2, x3,...) and hence llm has a surjective graph.	□
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Next example shows that the function $ induced by (^1, <^2, <^3,...) need not have a surjective graph if (^1, <^2, <^3,...) is not a morphism of ICU, even
if each /¿, and gi has a surjective graph and if gi o = ^ o / holds true for any positive integer i but i = 1.
Example 4.8. Let for each positive integer i and j > 1, X = Yj = [0,1], / = gj = ^ = 1[o,1], and let g1 : [0,1] ^ 2[0'1] be defined by its graph: r (g1) = [0,1] x [0,1]. Then the function $ induced by	• • •) does
not have a surjective graph.

Proof. Obviously, gi o = ^ o / holds true for any positive integer i > 1, and o/1(t) C g1 o<^2(t) for any t G [0,1]. Therefore <^2, <£3, • • •) induces $ defined by (1) according to Theorem 4.3.
Obviously (0,1,1,1, • • •) G jm{Yn,gn}~1 and lim{X„, /„}£= = {(t,t,t, • • •) | t G [0,1]}. But $(t, t, t, • • •) = {(t,t,t, •••)}, and therefore (0,1,1,1,•••) G $(t, t, t, • • •) for any t G [0,1].	□
In the rest of the section we study the transformation F : ZCU ^ CU, defined by
{Xn /«^1 1-> llim{Xn, /n}^=1
(^1, ^2, ^3, • • •) -A Hm •
CM
In Theorem 4.9 we show that the transformation F is very close to being a functor from ZCU to CU. Example 4.10 follows after the theorem to show CM	that F is not a functor from ZCU to CU.
Theorem 4.9. Let {Xra,/ra}£=1, {Y„,gn}£°=1 and {Z„, h„}£°=1 be any objects of ZCU, and
P = (^1,^2, ^3, • • •) : {Xn,fn}~=1 ^ ^ gn}~=1
and
^ = (^1,^3, •••) : {Y„,g„}~=1 ^ {Z„,h„}n=1 its morphisms. Then
1.	F(1Xi, 1X2 , 1X3 , • • •) = 1ljm{Xn,/n}~=1 ;
2.	(Fo F(p))(x) C F(^)(x) for all x G WX„, /„}~=1.
Proof. To prove (1), choose arbitrary x = (x1,x2,x3, • • •) G ljin{Xn, /n}^c=1. Then
F(1Xi, 1x2 , 1X3 , • • •) (x) = (1xi (x1 )x1x2 (x2)x1x3 (x3) X • • •)nlim{X„, /n}~=1
a	1
{x} = 1lmm{Xn)/n}~=i (x)
To prove (2), let x G hm{Xn, fn}n=1 and let
(N	x-
z G (Fty) o F(<))(x) = Fty)[F(<)(x)] = |J Fty)(y)
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be arbitrary. Then
z G	U	(^1(y1) x ^2) x--- ) n l^{Zn,hn}n=1
ye(^l(xi )x^2(x2)x---)nljm{Yn,gn }£°=1
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and therefore there is a point y G hm{Yra, gn}n=1 such that yn G <n(xn) and zn G ^n(yn) for each positive n. It follows that zn G |J(Xn) ^n(t) = ■0n[<n(xn)] = (^no<n)(xn) for each positive integer n and hence z G F(^D^)(x).
6 D F is a functor if and only if (Fo F(<))(x) = F(^D^)(x) holds

true for all x G hm{Xn, fn}n=1 and all objects {Xn,fn}n=1, {Yn,gn}n=1 and {Zn,hn}n=1 and all morphisms < = (<1, <2, <£3, • • •) : {Xn, fn}n=1 ^ {Yn,gn}n=1 and ^ = (^1,^2,^3, ...) : {Y*,gn}~1 ^ {Zn,hn}n=1 of ICU. Example 4.10 shows that this is not the case, hence F is not a functor.
1
Example 4.10. We use the notation from Theorem 4.9. Let for each positive integer n, Xn = Yn = Zn = [0,1] and let f,g : [0,1] ^ 2[0-1] be u.s.c. functions defined by f (t) = {t} and g(t) = [0,1] for each t G [0,1]. Also let f1 = h = = <n = g for each n > 2 and let <1 = fn+1 = gn = hn+1 = ^n+1 = f for each n > 1. Let x = (1, 0, 0, 0,...) G Xm{Xn,fn}n=1. Then (Fo F(<))(x) = F(^D<)(x).
CO
Proof. Let z = (1, 0, 0, 0,...) G hm{Zn, M^U. Obviously z G F(^D<)(x). Then, since <1(t) = {t} for each t G [0,1], y = (1,1,1,...) is the only element in hm{Yra,gn}n=1 such that y G F(<)(x). But, since F(^)(y) = [0,1] x {1} x {1} x ■ ■ ■ and z2 = 0 it follows that z G F(^)(y). Therefore z G (Fo F(<))(x) and hence (Fo F(<))(x) = F(^D<)(x).	□
l-H
5 An application
1
In the final section we study the following diagram.
CO
1 1	.	.
Theorem 5.1. Let Xj be compact metric spaces, and let fj : Xi+1 ^ 2Xi, gj : Xi+1 ^ 2Xi be u.s.c. functions, for all positive integers i and j. Let also for each j
L = Xm{Xj ,fj }i=1
0X11 CM 1
g1


00
g3
o
o» o
CM
i
cm 00 cm cm
£ CO CO
K1
CO CD
Jh
CD CO
u
a
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/11
X21
g1
/21
X31
g1
/31
X1
g!
/1
L1
/12
X2
gl
/22
X2
gl
/32
X2
g42
/42
L2
/13
X3
g3
/1
/23
X3
g3
/33
X3
g43
/43
L3
L
K2
K3
/2
and for each i
/3
K
K, = lim{Xj ,gj
If for each integer n, /n is the function induced by (/n, /n, /, the function induced by (gn, gn, gn, • • •), then
3, n,
.) and gn is
L
Jim{Lj ,gj }f=1
and
K = l^m{Kj,/j}tc=1.
are homeomorphic.
Proof. Define the function H : K — L as follows:
123
| <"Y» rf rP I X 1 . >Xj 1 • X
11
H(( x1
((x1, x2, x3
123
<"Y* rf rf x2, x2, x2>
222
I <"Y* i7> .7» 1, ^^
123
<"y» IP rY*
x3, x3, x3, 333 123
(3)
where (x1, x2, x3,...) G K, and (x1, x2, x3,...) G /, (x^, x2^, x3^,. for each positive integer i.
We will prove that H is a homeomorphism.
First, we prove that H is well-defined. We need to show that the right side of (3) is a point of L. The proof is in the following steps.
1
g
2
g
1.	x3, • • •) G Lj, for arbitrary j G N;
2.	x3, • • •) G gj (x^1, x2+1, xi+1, • • •), for arbitrary j G N
IN i—l
Let us prove (1).
Since (x1,x2,x3,. • •) G /-j (x^ x^ x3+1, • • •) = (lj /j(x^)) n Xj, it
follows that xj G /j (xj+1) for each i and j. Hence, (xj, x2, x3, • • •) G Lj. It remains to prove (2).
Since (x1, x2, x3, • • •) G Kj; it follows that for each i and j xj G gj(xj^1). Therefore (x!,x2,x3, • • •) G (nC=1 gj(xj)) n Lj = gj (x!+1, x2+1,x3+1, • • •) for all j.
Hence, ((x1, x^, x3, • • •), (x1, x2, x3, • • •) , (xf, x2, xj], • • •), • • •) G L. So we have proved that H : K ^ L is well defined.
In the same manner we prove that H' : L ^ K defined by
H ( (x1, x2, x3, • • •) , (xl, x2, x2, • • •) , (xl, x2, x3, • • •) , • • •) =
1 2 3	1 2 3	1 2 3
x1 , X^ • • • I , IX2,X2,X2,..•I , I x3, x3 , x3
CD U
((x1, x1, x1, • • •) , (x2, x2, x2, • • •)
is well defined. Since obviously H and H' are both continuous and inverses to each other, it follows that they are homeomorphisms.	□
00
Corollary 5.2. We use the notation of Theorem 5.1. If for all positive integers i and j
gjj o /jj+1 = /j o gj+1;
then the spaces L and K are homeomorphic.
Proof. The claim follows by Theorem 5.1 since by Corollary 4.4 there are induced functions /n and gn for each n.	□
We conclude the paper with the following example.
Example 5.3. Let X be any compact metric space and let / : X ^ X be a surjective single-valued mapping. Let L' = Hm{X, /-1}CC=17 where /-1 is the
1	X	^
u.s.c. function / : X ^ 2X defined by its graph
CD
r(/-1) = {(x,y) G X x X | (y,x) G r(/)}•
CD
Let a : L' ^ L' be the shift map, defined by
a(t1, ¿2, ¿3, • • •) = (¿2, ¿3, ¿4, • • •)
$H
for each (¿1, ¿2, ¿3, • • •) G L'.
Then the inverse limit ^m{L', a}CC=1 is homeomorphic to ^m{X, /}CC=1.
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Proof. We show first that the mapping (t1, t2, t3,...) M- {^(t1, t2, t3,...)} can be interpreted as an induced function and then we use Theorem 5.1 to prove that the inverse limit 1im{L', a}^^ is homeomorphic to 1im{X, /}n=1.
We use the notation that is used in Theorem 5.1. Let for all positive integers i, j, Xj = X, gj(t) = {/(t)}, and /j(t) = /-1(t) for each t G X.
Thengn(t1,t2,t3,...) = ({/(t1)}x{/(t2)}x{/(t3)}x.. .)nL' = {(t2,ta,t4,...)} {a(t1,t2,t3,...)} for any (t1,t2, t3,...) G L'. It follows that L =	gn}n=1 =
Jm{L>}n=1.
Let K' = Kn = 11m{X, /}n=1 for each positive integer n. Next we show that K = 1m{Kn, /n}n=1 = 11m{K', a'-1}n=1, where a' is the shift map from K' to K'. Note that a' is a homeomorphism, since / is single-valued, and that a'-1(t1,t2,t3,...) = (/(t1),t1,t2,t3,...) for each (t1, t2, t3,...) G K'.
Then /n(t1,t2, t3,...) = ({/-1(t1)} x {/-1(t2)} x {/-1(t3)} x ...) n K' = {(/(t1),t1,t2 ,t3,...)} = {a'-1(t1,t2,t3,...)} for any (t1 ,t2,t3,...) G K'. It follows that K = 11m{Kn, /„}£= = 1^{K', a'-1}n=1.
Since a'-1 is a homeomorphism it follows that K = 1im{K',a'-1}n=1 is homeomorphic to K' = 1im{X, /}n=1. By Theorem 5.1 Kls homeomorphic
to L, and that proves that 1im{X, /}n=1 is homeomorphic to 1jm{L', a}n=1.
1 1 □
i
Acknowledgements
This work was supported in part by the Slovenian Research Agency, under
Grants P1-0285 and P1-0297.
We thank the anonymous referee for his remarks and suggestions that S	helped us to correct several mistakes in the original version of the paper and
to improve its presentation.
References
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[2]	I. Banic, Inverse limits as limits with respect to the Hausdorff metric, Bull. Austral. Math. Soc. 75 (2007), 17-22.
[3]	I. Banic, Continua with kernels, Houston J. Math. 34 (2008), 145-163.
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IN	[20] V. Nall, Inverse limits with set valued functions, Houston J. Math. 37
(2011), 1323-1332.
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o
[24]	S. Varagona, Inverse limits with upper semi-continuous bonding functions and indecomposability, Houston J. Math. 37 (2011), 1017-1034.
Authors:
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Iztok Banič,
(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
iztok.banic@uni-mb.si
£
CO	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 matevz.crepnjak@um.si
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Goran Erceg,
Faculty of Natural Sciences and Mathematics, University of Split, Teslina 12, Split 21000, Croatia gorerc@pmfst.hr
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Matej Merhar,
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia
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matej.merhar@uni-mb.si
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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 uros.milutinovic@uni-mb.si
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