IMFM Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia Preprint series Vol. 51 (2013), 1188 ISSN 2232-2094 INVERSE LIMITS IN THE CATEGORY OF COMPACT HAUSDORFF SPACES AND UPPER SEMICONTINUOUS FUNCTIONS Iztok Banic Tina Sovic Ljubljana, May 17, 2013 !-h Inverse limits in the category of compact Hausdorff spaces and upper semicontinuous functions oo Iztok Banič, Tina Sovič O Ö CO CO CD ■ I u CD U University of Maribor, Slovenia Abstract We investigate inverse limits in the category CHU of compact Hausdorff spaces with upper semicontinuous (usc) functions. We introduce the notion of weak inverse limits in this category and show that the inverse limits with upper semicontinuous set-valued bonding functions (as they were defined in [15]) together with the projections are not necessarily inverse limits in CHU but they are always weak inverse limits in this category. This is a realization of our categorical approach to solving a problem stated by W. T. Ingram in [14]. £ CO Keywords: Upper semi-continuous functions, Inverse limits, Weak inverse limits 2000 Mathematics Subject Classification: primary 54C60; secondary 54B30 1 Introduction W. T. Ingram in his book [14] states the following problem: Problem 6.63. What can be said about inverse limits with set-valued functions if the underlying directed set is not a sequence of integers? CO In this paper we present a categorical approach to solving the above problem. Consider an inverse system (A, {Xa}aeÄ, {faß}a,ßeÄ) of compact Hausdorff spaces and continuous bonding functions. It is a well-known fact that the space l£m(A, {Xa }a€Ä, {faß }a,߀Ä) = 1—1 {(x7)7€a G TT Xa | for all a,ß G A, a < ß,xa = faß(xß)} 7eÄ together with the projection mappings p7 : Jim(A, {Xa} a€Ä) {faß }a,߀Ä) ^ XY, pY((xa)aeÄ) = xY, is in fact an inverse limit in the category CHC of compact Hausdorff spaces with continuous functions. In present paper we extend the category CHC to the category CHU of compact Hausdorff spaces with usc functions in such a way that CHC is interpreted as a proper subcategory of CHU. This can be done since every continuous function between compact Hausdorff spaces can be interpreted as 00 a usc function. As one of our main results we show that the inverse limits with upper semicontinuous set-valued bonding functions Jim(A, {Xa}a€Ä, {faß}a,߀Ä) = {(x7 )7€a G Yl Xa | for all a,ß G A, a < ß,xa G faß (xß)} together with the projections i Py : lim(A, {Xa}a€Ä) {faß}a,߀Ä) ^ Xy; 00 1 p7 ((xa)a€Ä ) = {x7 }, CSI are not necessarily inverse limits in the category but they are always so called weak inverse limits in CHU. CO In the second section we give the basic definitions that are used in the paper. In the third section we give a detailed description of the category CHU of compact Hausdorff spaces with usc bonding functions. In the fourth section we give results about inverse limits in the category CHU. In the last section we define objects in category CHU that are called weak inverse limits in this category. We also show that for any inverse system (A, {Xa}aeÄ, {faß}a,ßeÄ) in CHU, the corresponding inverse limit with upper semicontinuous set-valued bonding functions together with projections is always a weak inverse limit in category CHU. m CO CO CD 2 Definitions and notation i For any category K the class of objects of K will be denoted by Ob(K), the class of morphisms of K by Mor(K), and the partial binary associative CD U o CM IN 00 00 0 o CM 1 cm 00 cm cm £ CO CO operation (composition of morphisms) by o. For any X G Ob(K) the identity morphism on X will be denoted by 1X : X ^ X. For a directed set A (A is nonempty and equipped with a reflexive and transitive binary relation < with the property that every pair of elements has an upper bound), a family of objects {Xa | a G A} of K, and a family of morphisms {fap : Xp ^ Xa | a, ft G A, a < ft} of K, such that 1. for each a G A, faa = 1xa, 2. for each a, ft, 7 G A, from a < ft < 7 it follows that fap o fpY = faY, we call an inverse system (in K) and denote it by (A, {Xa}aeA, {fap }a,PeA)- We assume throughout the paper that A is cofinite, i.e. every a G A has at most finitely many predecessors. Next we define inverse limits in K. Definition 2.1. An object X G Ob(K), together with morphisms {pa : X ^ Xa | a G A} is an inverse limit of an inverse system (A, {Xa }aeA, {fap }«,peA) in the category K, if 1. for all a, ft G A, from a < ft it follows that the diagram, X p Xr Xp commutes; m CD $H CD m u a CD U 2. for any object Y G K and any family of morphisms {pa : Y ^ Xa | a G A} it follows that if the diagram Xa- faP Xp o CM IN 00 00 commutes, then there is a unique morphism p : Y ^ X such that for each a G A the diagram, Y P P X Xa 0 o CM 1 cm 00 cm cm £ CO CO m CD $H CD m u a CD U commutes. A map or mapping is a continuous function. If X is a compact Hausdorff space, then 2X denotes the set of all nonempty closed subsets of X. The graph Y(f) of a function f : X ^ 2Y is the set of all points (x,y) G X x Y such that y G f (x). A function f : X ^ 2Y is upper semi-continuous function if for each x G X and for each open set U C Y such that f (x) C U there is an open set V in X such that 1. x G V; 2. for all v G V it holds that f (v) C U. The following is a well-known characterization of usc functions between Hausdorff compacta (see [15, p. 120, Theorem 2.1]). Theorem 2.2. Let X and Y be compact Hausdorff spaces and f : X ^ 2Y a function. Then f is usc if and only if its graph T(f) is closed in X x Y. At the end of this section we introduce the notion of inverse limits with usc set-valued bonding functions as it was introduced by Mahavier in [19] and Ingram and Mahavier in [15]. In the last section we use this notion as a motivation for defining inverse limits with usc set-valued bonding functions for arbitrary inverse systems. An inverse sequence of compact Hausdorff spaces Xk with usc bonding functions fk is a sequence {Xk, fkwhere fk : Xk+i ^ 2Xk is usc for each k. The inverse limit with usc set-valued bonding functions of an inverse sequence {Xk, fk}£=1 is defined to be the subspace of the product space nr=i Xfc of all x = (X1,X2,X3,...) G rifcli Xk, such that xk G fk (xk+i) for IN 00 o Ö Gï CO -ifc=r .....—, ... each k. The inverse limit of {Xk, fkis denoted by ^im{Xk, fk}£=1. 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, 8, 10, 12, 16, 17, 18, 22, 23, 24, 25, 26]. 3 The category CHU The category CHU of compact Hausdorff spaces and usc functions consists of the following objects and morphisms: 1. Ob(CHU): compact Hausdorff spaces; 2. Mor(CHU): the usc functions from X to Y is the set of morphisms from X to Y, denoted by Mor (CHU)(X, Y). We also define the partial binary operation o (the composition) as follows. For each f G M or (CHU )(X,Y ) and each g G M or (CHU )(Y,Z ) we define g o f g M or (CHU )(X,Z ) by I CO for each x G X. £ Theorem 3.1. CHU is a category. (g ◦ f)(x) = g(f(x)) = U g(y) ye/(x) Proof. First we show that o is well-defined. Let f : X ^ Y and g : Y ^ Z be any morphisms. Let also x G X be arbitrary and let U be an open set in Z such that (g o f )(x) C U. Since g is usc and f (x) C Y, it holds that for each y G f (x) there is an open set Wy in Y such that 1. y G Wy; 2. for all w G Wy it holds that g(w) C U. y yv ; - Let W = Uyef(x) Wy. Since W is open in Y, f (x) C W, and since f is usc, Jy ' -- -- - - there is an open set V in X such that 1. x G V ; 2. for all v G V it holds that f (v) Ç W. a CD Let v £ V be arbitrary. Then (g ◦ f)(v) = g(f (v))= (J g(z) C U zef (v) since for each z £ f (v), it holds that g(z) C U. Therefore o is well-defined. It is obvious that the composition o of usc functions is an associative operation. All that is left to show is that for each X £ Ob(CWJ) 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 identity map 1X : X ^ X, defined by 1X (x) = {x} for each x £ X, is the usc function satisfying the above conditions. □ o 4 Inverse limits in CHU o CSF i CSF 00 CO CO CD u In this section we show that if (A, |Xa}aeA, {faß}a,ßeA) is an inverse system of compact Hausdorff spaces and usc set-valued bonding functions, then Jim(A, {Xa}«eA, {faß}a,߀Ä) (see Definition 4.1) together with the projections is not necessarily an inverse limit in the category CHU. Motivated by [15, 19], we define in Definition 4.1 objects in CHU, that are called inverse limits with usc set-valued bonding functions. Since such object were first introduced by Mahavier in [19] and Ingram and Mahavier in [15], where they call them the inverse limits with usc set-valued bonding functions, we continue to use the same name for them. Definition 4.1. Let (A, {Xa}aeÄ, {faß}a,ßeÄ) be any inverse system in CHU. We call the object l^m(A, {Xa}aeÄ, {faß Wsä) = {x £ JJ Xa | for all a < ß, Xa £ faß(xß)} a€Ä an inverse limit with usc set-valued bonding functions. CD In the following theorem we prove that ^—(A, {Xa}aeA, {faß Wsa) is really an object of CHU. CO Theorem 4.2. Let (A, {Xa}aeÄ, {faß}a;߀Ä) be any inverse system in CHU. Then the inverse limit with usc set-valued bonding functions ll—(A, {Xa}a€Ä, {faß Wsä) is a compact Hausdorff space. Proof. For each 7 G A, XY is a compact Hausdorff space, therefore the prod-uctn7eA XY is a compact Hausdorff space. Since ^im(A, {Xa}aeA, {faß}a)ßeA) is a subspace of the Hausdorff space, it is also a Hausdorff space. We show that ^im(A, {Xa}aeA, {faßis a closed subset of the compact space n7eA XY to show that it is compact. Let for all a, ß G A, a < ß, CSF r IN 00 00 £ CO CO CO CD ■ I J-H CD J-H u cu Gaß = r(faß) X Yl X7 = {x G JJ X7 | Xa G faß(Xß)}. 7€Ä\(a,ß} 7SÄ Since the graph r(faß) of faß is by Theorem 2.2 a closed subset of Xß X Xa Gaß is also a closed subset of n7eÄ XY. It is obvious that ]¿m(A, {X«} a a,ßeA,a<ß and hence l¿m(A, {Xa}aeA, {faß}a,ßeA) is a a closed subset of n7eA X7. □ In the following example we construct an inverse limit with usc set-valued bonding functions that is not an inverse limit in CHU regardless of the choice of morphisms {pa : X ^ Xa | a G A}. CO Example 4.3. Let A = N, Xk = [0,1], and let fk(k+1) = f for each k G N, where f : [0,1] ^ 2[0'1] is the function on [0,1] defined by its graph r(f) = {(t, t) G [0, 1] X [0, 1] | t G [0, 1]} U ({1} X [0, 1]). Also let X = hm(N, {[0,1]}keN, {fn}MeN) and let {p* : X ^ X, | i G N} be any set of morphisms in CHU, such that the diagrams (1) always commute. We show that X with {p* : X ^ X, | i G N} is not an inverse limit of (N, {[0,1]}keN, {fn}k/eN) in CHU. Let Y = [0,1] be an object in CHU and let {^k : Y ^ Xk | k G N} be the family of morphisms where (t) = [0,1] for each k and each t G Y. The diagram (2) always commutes. We distinguish the following two cases. 1. If there is a positive integer i0, such that 1 / pi0 (x) for each x G X, then suppose that $ is any morphism Y ^ X. Then (t) = [0,1] but 1 / pi0 ($(t)) for any t G Y. Therefore the diagram (3) does not commute for a = i0. 2. If for each positive integer i there is x* G X such that 1 G pi(xi), then let s G X be an accumulation point of the sequence {xi}°=1. We show CD first that pi(s) = [0,1] for each i. Let k be any positive integer. Then for each i > k, it follows from IN 00 00 c CD CO u [0,1] D pk(xe) = fki(j>i(xe)) D fke(1) D [0,1] that pk(xe) = [0,1]. Let |ni}°=1 be any increasing sequence of positive integers such that • ni > k for each i; • lim xni = s. It follows from pk (xni) = [0,1] that {xni} x [0,1] C r(pk) for each i. This means that for each t G [0,1], the point (xni , t) G r(pk). Therefore lim(xni,t) = (s,t) G r(pk) for each t, since r(pk) is a closed subset of X x [0,1]. It follows that {s} x [0,1] C r(pk) and hence pk(s) = [0,1]. Next, let $, ^ : Y ^ X be the morphisms in CHU, defined by & $(t) = X, CM I CM for each t G Y. It follows from cm *(t) = {s} n pk ($(t)) = pk (X ) = [0,1] = < (t) d and pk (*(t))= pk ({s}) = [0,1] = < (t) l-H that the diagram (3) commutes for both < = $ and < = Therefore there is no unique morphism < such that all diagrams (3) commute. Note that in the second part of Example 4.3, C $(t) = (UT=1 a, since ta £ Va (y) = faß (Vß (y)), there is tß £ ^ß (y), such that ta £ faß (tß). We continue inductively in the same fashion and choose for each i = k + 1,k + 2,k + 3,... and each ß £ Aj+1 an element tß £ <£a(y) such that ta £ faß(tß) for each a £ Aj, such that a < ß. CO Let x £ n 1&A XY be such an element that pY (x) = {tY} for each 7 £ A. It follows from the construction of x that x £ ^(y) and z £ pa(x). 5. Suppose that : Y ^ X is a morphism in CHU such that for each CO a £ A and for each y £ Y, pa(^(y)) = <£a(y). Let y £ Y be arbitrary CO and let z £ ^(y). Obviously z £ X since is a morphism from Y to X. It follows from pa(z) C pa(^(y)) = <£a(y) (for each a) that z £ EI7eA (y). Therefore z £ ^(y) and hence ^(y) C ^(y). 5 Weak inverse limits in CHU CD In this section we introduce the notion of weak inverse limits in CHU and show that lim(A, {Xa}aeA, {faß}a,ßeA) (together with the projections) is always a weakinverse limit in CHU. In Definition 5.1 we define a weak commutation of a diagram in the category CHU. Jh Definition 5.1. Let X,Y,Z £ Ob(CHU) and let f : X ^ Y, g : X ^ Z and h : Z ^ Y be any morphisms in CHU. The diagram o CM IN 00 00 0 o CM 1 cm 00 cm cm £ CO CO CO CD $H CD CO $H a CD U h Z weakly commutes, if for any x G X, f (x) C (h o g)(x). Example 5.2. Let f : [0,1] ^ 2[0>1], g : [0,1] ^ 2[0-1] be identity functions on [0,1] and let h : [0,1] ^ 2[0-1] be defined by h(x) = [0,1] for all x G [0,1]. Then the diagram, [0,1] [0,1] h [0,1] weakly commutes but does not commute. In the following definition we generalize the notion of inverse limits in CHU. Definition 5.3. An object X G Ob(CHU), together with morphisms {pa : X ^ Xa | a G A}, is a weak inverse limit of an inverse system, (A, {Xa}aeA, {fa/3}a,^eA) in CHU, if 1. for all a,^ G A, from a < ^ it follows that the diagram (1) weakly commutes; 2. for any object Y G CHU and any family of morphisms {^a : Y ^ Xa | a G A} it follows that if the diagram (2) commutes, then for any morphism ^ : Y ^ X such that for each a G A and for each y G Y, Pa(*(y)) = ^a (y), tf(y) C ([]^ (y)) n X holds true for all y G Y. Note that each inverse limit in CHU is always a weak inverse limit in CHU. IN Example 5.4. Let X = hm(N, {[0,1]}keN, {fke}k,em) be the inverse limit with usc set-valued bonding functions that we defined in Example 4.3. Then X, together with the projection mappings, is obviously not an inverse limit but it is a weak inverse limit in CHU. We show in the following theorem that the inverse limits with upper semicontinuous set-valued bonding functions together with projections are 00 always weak inverse limits in CHU. o Ö ¡5 CO CO Theorem 5.5. Let (A, {Xa}aeÄ, {faß}a;ßeÄ) be any inverse system in CHU. Then the inverse limit with usc set-valued bonding functions ^im(A, {Xa}a£A, {faß}a,߀Ä), together with projections Py : ]im(A, {Xa}aeÄ, {faß}a,߀Ä) ^ X7, I p7 ((Xa)aeA ) = }, 00 is a weak inverse limit of the inverse system (A, {Xa}aeÄ, {faß}a,ßeÄ) in CHU. Proof. Let X = l^im(A, {Xa}aeA, {faß Wsä)- First we prove that the diagram (1) weakly commutes. Choose any x G X and let a < ß. Then Pa(x) = {xa} C faß ({xß}) = (faß ◦ Pß)(x). Next, suppose that for an object Y G CHU and a family of morphisms {pa : Y ^ Xa | a G A} the diagram (2) commutes. By Lemma 4.4, for any morphism ^ : Y ^ X such that for each a G A and for each y G Y, Pa(^(y)) = Pa(y), tf(y) C (Ü7€a Py (y)) n X holds true for all y G Y. □ Acknowledgments CD The authors thank Uros Milutinovic for constructive discussion. This work was supported in part by the Slovenian Research Agency, under Grant P1-0285. Jh a CD Jh cu 00 1—1 References in 00 00 CO CD $H CD CO u u cu [1] I. Banic, On dimension of inverse limits with upper semicontinuous set-valued bonding functions, Topology Appl. 154 (2007), 2771-2778. [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. [4] I. Banic, M. Crepnjak, M. Merhar and U. Milutinovic, Limits of inverse limits, Topology Appl. 157 (2010), 439-450. [5] I. Banič, M. Črepnjak, M. Merhar and U. Milutinovic, Paths through inverse limits, Topology Appl. 158 (2011), 1099-1112. fi [6] I. Banič, M. Črepnjak, M. Merhar and U. Milutinovic, Towards the complete classification of generalized tent maps inverse limits , g http://dx.doi.org/l0.l0l6/j.topol-20l2.09.017. [7] I. Banič, M. Črepnjak, M. Merhar, U. Milutinovic and T. Sovič Waiewski's universal dendrite as an inverse limit with one set-valued bonding function, preprint (2012). [8] W. J. Charatonik and R. P. Roe, Inverse limits of continua having trivial shape, to appear in Houston J. Math. CO [9] W. J. Charatonik and R. P. Roe, Mappings between inverse limits of continua with multivalued bonding functions, preprint 10] A. N. Cornelius, Weak crossovers and inverse limits of set-valued functions , preprint (2009), 11] J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, London, Sydney, Toronto, 1966. 12] A. Illanes, A circle is not the generalized inverse limit of a subset of [0,1]2, Proc. Amer. Math. Soc. 139 (2011), 2987-2993. 13] A. Illanes, S. B. Nadler, Hyperspaces. Fundamentals and recent advances , Marcel Dekker, Inc., New York, 1999. 14] W. T. Ingram, An Introduction to Inverse Limits with Set-valued Functions , Springer, New York et al., 2012. CD o IN 00 00 0 G o 1 00 £ CO CO CO CD $H CD CO $H a CD U 15] W. T. Ingram, W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math. 32 (2006), 119-130. 16] W. T. Ingram, Inverse limits of upper semicontinuous functions that are unions of mappings, Topology Proc. 34 (2009), 17-26. 17] W. T. Ingram, Inverse limits with upper semicontinuous bonding functions: problems and some partial solutions, Topology Proc. 36 (2010), 353-373. 18] J. A. Kennedy and S. Greenwood, Pseudoarcs and generalized inverse limits, preprint (2010). 19] W. S. Mahavier, Inverse limits with subsets of [0,1] x [0,1], Topology Appl. 141 (2004), 225-231. 20] S. Mardešic and J. Segal, Shape theory, North-Holland, Amsterdam, 1982. 21] S. B. Nadler, Continuum theory. An introduction, Marcel Dekker, Inc., New York, 1992. 22] V. Nall, Inverse limits with set valued functions, Houston J. Math. 37 (2011), 1323-1332. 23] V. Nall, Connected inverse limits with set valued functions, Topology Proc. 40 (2012), 167-177. 24] V. Nall, Finite graphs that are inverse limits with a set valued function on [0,1], Topology Appl. 158 (2011), 1226-1233. 25] A. Palaez, Generalized inverse limits, Houston J. Math. 32 (2006), 11071119. 26] S. Varagona, Inverse limits with upper semi-continuous bonding functions and indecomposability, Houston J. Math. 37 (2011), 1017-1034. Authors: 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 00 1-H o CM IN Tina Sovič, Faculty of Civil Engineering, University of Maribor, Smetanova 17, Maribor 2000, Slovenia tina.sovic@um.si 00 00 0 G o CM 1 cm 00 cm cm £ CO CO CO CD $H CD CO u a CD U