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Section B:

b-8

Basic constructions

Inverse Systems and Direct Systems

1. Ordered sets A preordering (or quasi-order) on a set Λ is a binary relation on Λ which is reflexive, i.e., λ λ for each λ ∈ Λ, and transitive, i.e., λ λ and λ λ implies λ λ . The preordering is an ordering when it is antisymmetric, i.e., λ λ and λ λ imply λ = λ . A preordered set (Λ, ) is directed provided, for each λ , λ ∈ Λ, there is a λ ∈ Λ such that λ , λ λ. We say (Λ, ) is cofinite preordered set provided it is an ordering and each element λ ∈ Λ admits only finitely many predecessors, λ1 , λ2 , . . . , λn λ. The set N = (N, ) of all positive integers with the usual order and a singleton set {∗} are cofinite directed ordered sets. A function f : M → Λ between preordered sets M = (M, ) and Λ = (Λ, ) is increasing provided f (µ) f (µ ) for µ µ in M. 2. Direct systems and limits A direct system {Xλ , pλλ , Λ} of topological spaces consists of a directed ordered set Λ = (Λ, ), topological spaces Xλ for λ ∈ Λ, and continuous maps pλλ : Xλ → Xλ for λ λ (the bonding maps) satisfying the conditions pλλ = id for λ ∈ Λ and pλ λ pλλ = pλλ for λ λ λ . A direct sequence {Xi , pi,i+1 } of topological spaces is a direct system {Xi , pii , N} where pii = pi+k,i ◦ · · · ◦ pi+1,i+2 ◦ pi,i+1 for i = i + k + 1 and pii = id for i = i. Forany direct system {Xλ , pλλ , Λ}, consider the direct sum Xλ . Two elements xλ ∈ Xλ and xλ ∈ Xλ are said to be equivalent if pλλ (xλ ) = pλ λ (xλ ) for some λ λ, λ . This relation R defines an equivalence relation on Xλ , and the quotient space X = Xλ /R is called the direct limit (or limit, for short) of {Xλ , pλλ , Λ}. Then the natural projections pλ : Xλ → X for λ ∈ Λ satisfy pλ = pλ pλλ for λ λ . In a similar way, direct systems of sets, (Abelian) groups are defined, and their limits exist. 3. Inverse systems and limits An inverse system {Xλ , pλλ , Λ} of topological spaces consists of a directed ordered set Λ = (Λ, ), topological spaces Xλ for λ ∈ Λ, and continuous maps pλλ : Xλ → Xλ for λ λ (the bonding maps) satisfying the conditions pλλ = id for λ ∈ Λ and pλλ pλ λ = pλλ for λ λ λ .

An inverse sequence {Xi , pi,i+1 } of topological spaces is an inverse system {Xi , pii , N} where pii = pi,i+1 ◦ pi+1,i+2 ◦ · · · ◦ pi+k,i for i = i + k + 1 and pii = id for i = i. The inverse limit (or limit for short) of {Xλ , pλλ , Λ} is the subspace of the product λ∈Λ Xλ consisting of all points x = (xλ ) such that pλλ (xλ ) = xλ for λ < λ . Then the projections pλ : X → Xλ for λ ∈ Λ satisfy pλ = pλλ pλ for λ < λ . The limit X of an inverse system of (non-empty) compact Hausdorff spaces Xλ is a (non-empty) compact space. However, this is not true for noncompact spaces. For example, the inverse sequence consisting of the open intervals (0, 1/n) (n = 1, 2, . . .) and inclusions has an empty limit. In a similar way, inverse systems of sets, (Abelian) groups, compact spaces, topological groups are defined, and their limits exist as a set, (Abelian) group, compact space, topological space, respectively. However, the limit of an inverse system of compact polyhedra does not exist as a compact polyhedron in general. For example, there is a decreasing sequence of compact polyhedra whose intersection X is not a polyhedron. Such a sequence is viewed as an inverse sequence {Xi , pi,i+1 } with each pi,i+1 being an inclusion and its limit being X. Moreover, if we consider an inverse system {Xλ , pλλ , Λ} of topological spaces with each pλλ being homotopy classes, its limit may not exist at all ([14, p. 56], [4]). In general, we can define an inverse system and its limit for a category C as follows: An inverse system X = {Xλ , pλλ , Λ} in the category C indexed by Λ consists of a directed ordered set Λ = (Λ, ), of objects Xλ for λ ∈ Λ, and of morphisms pλλ : Xλ → Xλ for λ λ satisfying the conditions: pλλ = id for λ ∈ Λ and pλλ pλ λ = pλλ for λ λ λ . Before we define limits in C, we will introduce the notion of pro-category. A system morphism f = {f, fµ } : X → Y between inverse systems X = {Xλ , pλλ , Λ} and Y = {Yµ , qµµ , M} consists of a function f : M → Λ, of morphisms fµ : Xf (µ) → Yµ in the category C satisfying the condition For any µ, µ ∈ M with µ µ , there is a λ ∈ Λ such that f (µ), f (µ ) λ and fµ pf (µ)λ = qµµ fµ pf (µ )λ . If Z = {Zν , rνν , N} is an inverse system in C and if g = {g, gν } : Y → Z is a system morphism, then we define the composition of morphisms h = {h, hν } : X → Z of f and g as follows: h = fg : N → Λ and hν = gν fg(ν) :

b-8

Inverse systems and direct systems

Xh(ν) → Zν for each ν ∈ N . Thus the defined h is a system morphism, and the associativity of compositions holds. We can also define the identity morphism 1X = {1, 1Xλ } : X → X where 1 = 1Λ : Λ → Λ and 1Xλ : Xλ → Xλ are the identities. Thus, all inverse systems and system morphisms form a category inv-C. Now we define an equivalence relation ≡ on system morphisms as follows: Let f = {f , fµ } : X → Y be a system morphism. We say that f is congruent to f , in notation, f ≡ f , provided that for each µ ∈ M there is a λ ∈ Λ with f (µ), f (µ) λ and fµ pf (µ)λ = fµ pf (µ)λ . The equivalence class of the system morphism f is denoted by [f ]. It is easy to see that composition in inv-C induces a well-defined composition of equivalence classes. Thus, all inverse systems and all equivalence classes of system morphisms form a category pro-C, which is called the pro-category of C. In particular, if f : (X) → Y is a morphism of pro-C from a rudimentary inverse system, i.e., an inverse system indexed by a singleton (X), to an inverse system {Yµ , qµµ , M}, then it satisfies the property: (PROJ) fµ = qµµ fµ for µ µ . A limit of an inverse system X is a morphism p : X → X in pro-C for some object X of C with the universal property: (UP) For any morphism p : X → X in pro-C, there exists a unique morphism f : X → X of C such that pf = p , that is, pλ f = pλ for each λ ∈ Λ. Consequently, if a limit exists, it is unique up to natural isomorphism. We refer to an object X, denoted by lim X, and a morphism p : X → X as a limit and a natural projection, respectively. Every inverse system in C has a limit provided the following two conditions are satisfied: (L1) Every collection of objects has a product in C; and (L2) Every pair of morphisms f0 , f1 : X → Y has an equalizer, i.e., a morphism j : Y → Z such that (1) jf0 = jf1 ; and (2) If j : Y → Z is another morphism such that j f0 = j f1 , then there is a morphism h : Z → Z such that hj = j . In this case, any system morphism f : X → Y induces a morphism f , denoted by lim f : lim X → lim Y in C, such that qµ f = fµ pf (µ) for any µ, which defines a functor lim : pro-C → C. Let Top denote the category of topological spaces and continuous maps. The following are the full subcategories of Top that are often used in general topology: Notation

Objects

Top3.5 CTop3.5 CH CM

Tychonoff spaces Dieudonné complete Tychonoff spaces compact Hausdorff spaces compact metric spaces Absolute Neighborhood Retracts polyhedra with the CW-topology

ANR Pol

57 4. Approximate limits and approximate resolutions Compact spaces X are often studied by expressing X as the limit of an inverse system of compact polyhedra, whose idea goes back to the work of P.S. Alexandroff [1] (see also [5, 6, 9]). Indeed, every compact Hausdorff space X admits an inverse system X = {Xλ , pλλ , Λ} and an inverse limit p : X → X. Moreover, the limit of any inverse system of non-empty compact Hausdorff spaces is a non-empty compact space. However, the theory of inverse limits of nonmetric compact spaces has some defects. A classical result of H. Freudenthal [6] asserts that every compact metric space X with dim X = n is the limit of an inverse sequence of finite polyhedra of dimension n. However, this result cannot be generalized to compact Hausdorff spaces as pointed out in [10, 23]. S. Mardeši´c and L. R. Rubin [13] then introduced the notion of approximate inverse limits, and many theorems for compact metric spaces have been extended to compact Hausdorff spaces. For maps f, g from a topological space X to a metric space (X, d) and ε > 0, we write d(f, g) ε if d(f (x), g(x)) ε for each x ∈ X. An approximate inverse system X = {Xλ , ελ , pλλ , Λ} of compact metric spaces consists of a directed ordered set (Λ, ) with no maximal element, a compact metric space Xλ with the metric dλ and a real number ελ > 0 for each λ ∈ Λ, a continuous map pλλ : Xλ → Xλ for each pair λ λ , satisfying the following conditions (A0), (A1), (A2) and (A3): (A0) (∀λ ∈ Λ)(pλλ = id). (A1) (∀λ, ∀λ , ∀λ ∈ Λ) (λ λ λ =⇒ dλ (pλλ pλ λ , pλλ ) ελ ). (A2) (∀λ ∈ Λ)(∀δ > 0)(∃λ0 λ) (∀λ , ∀λ ∈ Λ) (λ0 λ λ =⇒ dλ (pλλ pλ λ , pλλ ) δ). (A3) (∀λ ∈ Λ)(∀δ > 0)(∃λ0 λ)(∀λ λ0 ) (∀x, ∀x ∈ Xλ ) (dλ (x, x ) ελ =⇒ dλ (pλλ (x), pλλ (x )) δ). The approximate limit (or limit, for short) of an approxi mate inverse system X = {X λ , ελ , pλλ , Λ} is defined to be the subspace of the product λ∈Λ Xλ consisting of all points x = (xλ ) satisfying that: (AL) (∀λ ∈ Λ)(∀δ > 0)(∃λ0 λ)(∀λ ∈ Λ) (λ λ0 =⇒ d(xλ, pλλ (xλ )) δ). S. Mardeši´c and L.R. Rubin [13] proved that a compact Hausdorff space X has dim X n if and only if it is the limit of an approximate inverse system of finite polyhedra, whose dimension n. On the other hand, without compactness, the concept of inverse limits leads to many pathological phenomena, and hence, inverse limits are not often used for noncompact spaces. Indeed, there is an inverse sequence of 0-dimensional paracompact spaces whose limit is a normal space with positive covering dimension [3]. S. Mardeši´c [11] then introduced the notion of resolution of topological spaces to deal with non-compact spaces, extending

58

Section B:

the properties of compact inverse limits. Let N (X) denote the set of all normal covers of X. For two maps f, g : X → Y and U ∈ N (Y ), we write (f, g) U if for each x ∈ X there exists U ∈ U such that {f (x), g(x)} ⊆ U . Let X be a topological space and X = {Xλ , pλλ , Λ} be a inverse system of topological spaces. Then a map of system f = {fλ , Λ} : X → X is defined to be a collection of continuous maps fλ : X → Xλ , λ ∈ Λ, such that pλλ fλ = fλ for each λ, λ ∈ Λ with λ λ . A resolution of X is a map of system f = (fλ , Λ) : X → X which satisfies the following conditions (R1) and (R2): (R1) If P is a polyhedron, U ∈ N (P ) and f : X → P is a continuous map, then there exist λ ∈ Λ and a continuous map g : Xλ → P such that (gfλ , f ) U . (R2) If P is a polyhedron, U ∈ N (P ), then there exists V ∈ N (P ) with the following property: For each λ ∈ Λ and each pair of two continuous maps h, h : Xλ → P , if (hfλ , h fλ ) V, then there exists λ λ such that (hpλλ , h pλλ ) U . Combining the notions of approximate limits and resolutions, S. Mardeši´c and T. Watanabe [16] introduced the notion of approximate resolutions to overcome the defects of inverse limits of non-compact and non-metric spaces. An approximate inverse system X = {Xλ , Uλ , pλλ , Λ} of topological spaces consists of a directed ordered set (Λ, ) with no maximal element, a topological space Xλ and Uλ ∈ N (Xλ ) for each λ ∈ Λ, a continuous map pλλ : Xλ → Xλ for each pair λ λ , satisfying the following conditions (B0), (B1), (B2) and (B3): (B0) (∀λ ∈ Λ)(pλλ = id). (B1) (∀λ, ∀λ , ∀λ ∈ Λ) (λ λ λ =⇒ (pλλ pλ λ , pλλ ) Uλ ). (B2) (∀λ ∈ Λ)(∀V ∈ N (Xλ ))(∃λ0 λ) (∀λ , ∀λ ∈ Λ) (λ0 λ λ =⇒ (pλλ pλ λ , pλλ ) V). (B3) (∀λ ∈ Λ)(∀V ∈ N (Xλ ))(∃λ0 λ) (∀λ λ0 ) −1 (Uλ is a refinement of {pλλ (U ): U ∈ V}). Let X be a topological space and let X = {Xλ , Uλ , pλλ , Λ} be an approximate inverse system of topological spaces. Then an approximate map f = {fλ , Λ} : X → X is defined to be a collection of continuous maps fλ : X → Xλ , λ ∈ Λ, satisfying: (AM) (∀λ ∈ Λ)(∀U ∈ N (Xλ ))(∃λ0 λ)(∀λ ∈ Λ) (λ λ0 =⇒ (pλλ fλ , fλ ) U). An approximate resolution of X is an approximate map f = {fλ , Λ} : X → X that satisfies the conditions (R1) and (R2) above. T. Watanabe [28] proved that a topological space X has dim X n if and only if it admits an approximate resolution consisting of polyhedra with dimension n. The above generalizations of inverse limits are summarized as in the following diagram: Inverse limits for CM ↓ Approximate limits for CH

−→ −→

Resolutions for Top ↓ Approximate resolutions for Top

Basic constructions

5. Further development Maps between topological spaces can be also studied by use of inverse systems and their modifications. A map f : X → Y between compact Hausdorff spaces admits compact polyhedral inverse systems X = {Xλ , pλλ , Λ}

and Y = {Yµ , qµµ , M}

and system morphism f = {g, fµ } : X → Y such that lim f = f [8], which means that (M) whenever µ µ , there is a λ > g(µ), g(µ ) such that fµ pg(µ)λ = qµµ fµ pg(µ )λ , and (LM) fµ pg(µ) = qµ f for µ ∈ M. An analogous fact for resolutions was obtained in [11]. However, if we choose the polyhedral inverse systems X and Y in Pol in advance, it may not be possible to find fµ with the strict commutativity conditions (M) and (LM) as pointed out in [10, 17, 25, 26]. Indeed, consider any map f from a Cantor set C onto the unit interval I , where the Cantor set C is the limit of an inverse sequence X of finite sets. Then there is no system morphism f : X → I , where I is a rudimentary inverse system, such that lim f = f since all terms of X are finite. If these conditions are replaced by approximate commutativity conditions, this will become possible. This was done for resolutions by T. Watanabe [26], for compact approximate limits by S. Mardeši´c and J. Segal [15], and for approximate resolutions by S. Mardeši´c and T. Watanabe [16]. In general, maps between approximate resolutions with some reasonable conditions can be defined by using approximate commutativity conditions. Moreover, one can obtain a category whose objects are those approximate resolutions and whose morphisms are reasonably defined equivalence classes of those maps. This category gives a useful tool in studying topological spaces and continuous maps since it is equivalent to the category CTop3.5 More generally, inverse systems and their generalizations give basic tools in many applications. S. Mardeši´c [11] used resolutions to extend shape theory over topological spaces. In fact, inverse systems, approximate inverse limits and approximate resolutions can be used to define shape theory in a categorical way for compact metric spaces, compact Hausdorff spaces, topological spaces, respectively. Similarly, such inverse system approach can also be used to ˇ define strong shape theory [2, 7, 12]. Moreover, Cech homology and Steenrod homology theories are also studied by using the pro-category pro-HPol of the homotopy category of polyhedra [27, 12]. The theory of approximate resolutions for uniform spaces and its applications in uniform spaces were obtained in [18]. The theory of approximate resolutions was also used to study fixed-point theory [24], and more recently, T. Miyata and T. Watanabe applied it to fractal geometry and obtained interesting results [19, 21].

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Inverse systems and direct systems

References [1] P.S. Alexandroff, Une définition des nombres de Betti pour un ensemble fermé quelconque, C. R. Acad. Sci. Paris 184 (1927), 317–319. [2] F. Cathey and J. Segal, Strong shape theory and resolutions, Topology Appl. 15 (1983), 119–130. [3] M.G. Charalambous, An example concerning inverse limit sequences of normal spaces, Proc. Amer. Math. Soc. 78 (1980), 605–608. [4] J.M. Cohen, The homotopy groups of inverse limits, Proc. London Math. Soc. 27 (1973), 159–192. [5] S. Eilenberg and N.E. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press, Princeton, NJ (1952). [6] H. Freudenthal, Entwicklungen von Räumen und ihren Gruppen, Compositio Math. 4 (1937), 145–234. [7] B. Günther, Approximate resolutions in strong shape theory, Glasnik Mat. Ser. III 29 (49) (1994), 109–122. [8] J. Haxhibeqiri, Shape fibrations for compact Hausdorff spaces, Publ. Inst. Math. Beograd (N.S.) 31 (45) (1982), 33–49. [9] S. Lefschetz, On compact spaces, Ann. of Math. 32 (1931), 521–538. [10] S. Mardeši´c, Mappings of inverse systems, Glasnik Mat. Fiz. Astr. Ser. II 18 (1960), 241–254. [11] S. Mardeši´c, Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math. 114 (1981), 53–78. [12] S. Mardeši´c, Strong Shape and Homology, Springer, Berlin (2000). [13] S. Mardeši´c and L.R. Rubin, Approximate inverse systems of compacta and covering dimension, Pacific J. Math. 138 (1989), 129–144. [14] S. Mardeši´c and J. Segal, Shape Theory, NorthHolland, Amsterdam (1982). [15] S. Mardeši´c and J. Segal, Mapping approximate inverse systems of compacta, Fund. Math. 134 (1990), 73–91.

59 [16] S. Mardeši´c and T. Watanabe, Approximate resolutions of spaces and mappings, Glasnik Mat. 24 (1989), 587– 637. [17] J. Mioduszewski, Mappings of inverse limits, Colloq. Math. 10 (1963), 39–44. [18] T. Miyata and T. Watanabe, Approximate resolutions of uniform spaces, Topology Appl. 113 (2001), 211–241. [19] T. Miyata and T. Watanabe, Lipschitz functions and approximate resolutions, Topology Appl. 122 (2002), 353–375. [20] T. Miyata and T. Watanabe, Approximate resolutions and boxcounting dimension, Topology Appl., submitted for publication. [21] T. Miyata and T. Watanabe, BiLipschitz maps and approximate resolutions, Glasik Mat. 38 (2003), 129– 155. [22] K. Morita, On shapes of topological spaces, Fund. Math. 85 (1975), 251–259. [23] B.A. Pasynkov, On polyhedral spectra and dimension of bicompacta and of bicompacta groups, Dokl. Akad. Nauk SSSR 121 (1958), 45–48. [24] J. Segal and T. Watanabe, Cosmic approximate limits and fixed points, Trans. Amer. Math. Soc. 333 (1992), 1–61. [25] T. Watanabe, Approximate expansions of maps into inverse systems, Geometric and Algebraic Topology, Banach Center Publ., Vol. 18 (1986), 363–370. [26] T. Watanabe, Approximative shape I–IV, Tsukuba J. Math. 11 (1987), 17–59, 303–339 and 12 (1988), 1– 41, 273–319. ˇ [27] T. Watanabe, Cech homology, Steenrod homology and strong homology I, Glasnik Mat. Ser. III 22 (1987), 187–238. [28] T. Watanabe, Approximate resolutions and covering dimension, Topology Appl. 38 (1991), 147–154.

Tadashi Watanabe Yamaguchi, Japan