() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 11, No. 2, 2010 pp. 95-115 Closed injective systems and its fundamental limit spaces Marcio Colombo Fenille Abstract. In this article we introduce the concept of limit space and fundamental limit space for the so-called closed injected systems of topological spaces. We present the main results on existence and uniqueness of limit spaces and several concrete examples. In the main section of the text, we show that the closed injective system can be con- sidered as objects of a category whose morphisms are the so-called cis- morphisms. Moreover, the transition to fundamental limit space can be considered a functor from this category into the category of topological spaces and continuous maps. Later, we show results about properties on functors and counter-functors for inductive closed injective system and fundamental limit spaces. We finish with the presentation of some results of characterization of fundamental limit spaces for some special systems and the study of the so-called perfect properties. 2000 AMS Classification: 18A05, 18A30, 18B30, 54A20, 54B17. Keywords: closed injective system, limit space, category, functoriality, com- patibility of limits, perfect property. 1. Introduction The purpose of this article is to introduce and study what we call the cat- egory of closed injective systems and cis-morphisms and the concept of limit spaces of such systems. We start by defining the so-called closed injective systems (CIS to shorten), and the concepts of limit space for such systems. We have particular interest in a special type of limit space, those we call fundamental limit space. In Section 3 we introduce this concept and we demonstrate theorems of existence and uniqueness of fundamental limit spaces. In Section 4 we present some very illustrative examples. 96 M. C. Fenille Section 5 is one of the most important and interesting for us. There we show that a closed injective system can be considered as a object of a category, whose morphisms are the so-called cis-morphisms, which we define in this occasion. Furthermore, we prove that this category is complete with respect to direct limits, that is, all inductive system of CIS’s and cis-morphisms has a direct limit. In Section 6, we prove that the transition to the fundamental limit can be considered as a functor from the category of CIS’s and cis-morphisms into the category of topological spaces and continuous maps. In Section 7, we show that the transition to the direct limit on the cate- gory of CIS’s and cis-morphisms is compatible (in a way) to transition to the fundamental limit space. In section 8, we study a class of special CIS’s called inductive closed injec- tive systems. In the two following sections, we study the action of functors and counter-functors, respectively, in such systems, and present some simple applications of the results demonstrated. We finish with the presentation of some results of characterization of funda- mental limit space for some special systems, the so-called finitely-semicomponi- ble and stationary systems, and the study of the so-called perfect properties over topological spaces of a system and over its fundamental limit spaces. 2. Closed injective systems and limit spaces Let {Xi} ∞ i=0 be a countable collection of nonempty topological spaces. For each i ∈ N, let Yi be a closed subspace of Xi. Assume that, for each i ∈ N, there exists a closed injective continuous map fi : Yi → Xi+1. This structure is called closed injective system, or CIS, to shorten. We write {Xi,Yi,fi} to represent this system. Moreover, by injection we mean a injective continuous map. We say that two injection fi and fi+1 are semicomponible if fi(Yi)∩Yi+1 6= ∅. In this case, we can define a new injection fi,i+1 : f −1 i (Yi+1) → Xi+2 by fi,i+1(y) = (fi+1 ◦ fi)(y), for all y ∈ f −1 i (Yi+1). For convenience, we put fi,i = fi. Moreover, we say that fi is always semicomponible with itself. Also, we write fi,i−1 to be the natural inclusion of Yi into Xi for each i ∈ N. Given i,j ∈ N, j > i + 1, we say that fi and fj are semicomponible if fi,k and fk+1 are semicomponible for all i + 1 ≤ k ≤ j − 1, where fi,k : f −1 i,k−1(Yk) → Xk+1 is defined inductively. To facilitate the notations, if fi and fj are semicom- ponible, we write Yi,j = f −1 i,j−1(Yj ), CIS’s and its fundamental limit spaces 97 that is, Yi,j is the domain of the injection fi,j . According to the agreement fi,i = fi, we have Yi,i = Yi. Lemma 2.1. If fi and fj are semicomponible, i < j, then fk and fl are semicomponible, for any integers k,l with i ≤ k ≤ l ≤ j. If fi and fj are not semicomponible, then fi and fk are not semicomponible, for any integers k > j. Lemma 2.2. If fi and fj are semicomponible, with i < j, then Yi,j = (fj−1 ◦ · · · ◦ fi) −1(Yj ) and fi,j (Yi,j ) = (fj ◦ fi,j−1)(Yi,j−1). The proofs of above results are omitted. Henceforth, since products of maps do not appear in this paper, we can sometimes omit the symbol ◦ in the composition of maps. Definition 2.3. Let {Xi,Yi,fi} be a CIS. A limit space for this system is a topological space X and a collection of continuous maps φi : Xi → X satisfying the conditions: L.1. X = ⋃∞ i=0 φi(Xi); L.2. Each φi : Xi → X is a imbedding; L.3. φi(Xi) ∩φj (Xj ) . = φjfi,j−1(Yi,j−1) if i < j and fi and fj are semicom- ponible; L.4. φi(Xi) ∩ φj (Xj ) = ∅ if fi and fj are not semicomponible; where . = indicates, besides the equality of sets, the following: If x ∈ φi(Xi) ∩ φj (Xj ), say x = φi(xi) = φj (xj ), with xi ∈ Xi and xj ∈ Xj , then we have necessarily xi ∈ Yi,j−1 and xj = fi,j−1(xi). Remark 2.4. The “pointwise identity” indicated by . = in L.3 reduced to iden- tity of sets indicates only that φi(Xi) ∩ φj (Xj ) = φi(Yi,j−1) ∩ φjfi,j−1(Yi,j−1). The existence of different interpretations of condition L.3 is very important. Furthermore, equivalent conditions to those of the definition can be very useful. The next results give us some practical interpretations and equivalences. Lemma 2.5. Let {X,φi} be a limit space for the CIS {Xi,Yi,fi} and suppose that fi and fj are semicomponible, i < j. Then φjfi,j−1(yi) = φi(yi) for yi ∈ Yi,j−1. Proof. Let yi ∈ Yi,j−1 be a point. By condition L.3 we have φjfi,j−1(yi) ∈ φi(Xi), that is, φjfi,j−1(yi) = φi(xi) for some xi ∈ Xi. Again by condition L.3, xi ∈ Yi,j−1 and fi,j−1(xi) = fi,j−1(yi). Since each fk is injective, also fi,j−1 is injective. Therefore xi = yi, which implies φjfi,j−1(yi) = φi(yi). � Lemma 2.6. Let {X,φi} be a limit space for the CIS {Xi,Yi,fi} and suppose that fi and fj are semicomponible, with i < j. Then φi(Xi − Yi,j−1) ∩ φj (Xj − fi,j−1(Yi,j−1)) = ∅. 98 M. C. Fenille Proof. It is obvious that if x ∈ φi(Xi − Yi,j−1) ∩ φj (Xj − fi,j−1(Yi,j−1)) then x ∈ φi(Xi) ∩ φj (Xj ) . = φjfi,j−1(Yi,j−1). But this is a contradiction, since φj is an imbedding, and so φj (Xj − fi,j−1(Yi,j−1)) = φj (Xj ) − φjfi,j−1(Yi,j−1). � Proposition 2.7. Let {Xi,Yi,fi} be an arbitrary CIS and let φi : Xi → X be imbedding into a topological space X = ∪∞i=0φi(Xi) satisfying the following properties: L.4. φi(Xi) ∩ φj (Xj ) = ∅ always that fi and fj are not semicomponible; L.5. φjfi,j−1(yi) = φi(yi) for every yi ∈ Yi,j−1, always that fi and fj are semicomponible, with i < j; L.6. φi(Xi − Yi,j−1) ∩ φj (Xj − fi,j−1(Yi,j−1)) = ∅, always that fi and fj are semicomponible, with i < j. Then {X,φi} is a limit space for the CIS {Xi,Yi,fi}. Proof. We will prove that condition L.3 is true. Suppose that fi and fj are semicomponible, with i < j. By condition L.5, the sets φi(Xi) ∩ φj (Xj ) and φjfi,j−1(Yi,j−1) are nonempty. We will prove that they are pointwise equal. Let x ∈ φi(Xi)∩φj (Xj ), say x = φi(xi) = φj (xj ) with xi ∈ Xi and xj ∈ Xj . Suppose, by contradiction, that xi /∈ Yi,j−1. Then φi(xi) ∈ φi(Xi − Yi,j−1). By condition L.6 we must have φj (xj ) = φi(xi) /∈ φj (Xj − fi,j−1(Yi,j−1)), that is, φj (xj ) ∈ φjfi,j−1(Yi,j−1). So xj ∈ fi,j−1(Yi,j−1). Thus, there is yi ∈ Yi,j−1 such that fi,j−1(yi) = xj . By condition L.5, φi(yi) = φjfi,j−1(yi) = φj (xj ). However, φj (xj ) = φi(xi). It follows that φi(yi) = φi(xi), and so xi = yi ∈ Yi,j−1, which is a contradiction. Therefore xi ∈ Yi,j−1. In order to prove the remaining, take x ∈ φi(Xi) ∩ φj (Xj ), x = φi(yi) = φj (xj ), with yi ∈ Yi,j−1 and xj ∈ Xj . We must prove that xj = fi,j−1(yi). By condition L.5, φjfi,j−1(yi) = φi(yi) = φj (xj ). Thus, the desired identity is obtained by injectivity. This proves that φi(Xi) ∩ φj (Xj ) . = φjfi,j−1(Yi,j−1) and so that {X,φi} is a limit space for {Xi,Yi,fi}. � Corollary 2.8. Condition L.3 can be replaced by both together L.5 and L.6. Proof. Lemmas 2.5 e 2.6 and Proposition 2.7 implies that. � Theorem 2.9. Let {Xi,Yi,fi} be a CIS. Assume that {X,φi} and {Z,ψi} are two limit spaces for this CIS. Then there is a unique bijection (not necessarily continuous) β : X → Z such that ψi = β ◦ φi for every i ∈ N. Proof. Define β : X → Z in the follow way: For each x ∈ X, we have x = φi(xi), for some xi ∈ Xi. Then, we define β(x) = ψi(xi). We have: • β is well defined. Let x ∈ X be a point with x = φi(xi) = φj (xj ), where xi ∈ Xi, xj ∈ Xj and i < j. Then x ∈ φi(Xi) ∩ φj (Xj ) . = φjfi,j−1(Yi,j−1) and xj = fi,j−1(xi) by condition L.3. Thus ψj (xj ) = ψjfi,j−1(xi) = ψi(xi), where the latter identity follows from condition L.3. • β is injective. Suppose that β(x) = β(y), x,y ∈ X. Consider x = φi(xi) and y = φj (yj ), xi ∈ Xi, yj ∈ Xj, i < j (the case where j < i is symmetrical and the case where i = j is trivial). Then ψi(xi) = β(x) = β(y) = ψj (yj ). CIS’s and its fundamental limit spaces 99 It follows that ψi(xi) = ψj (yj ) ∈ ψi(Xi) ∩ ψj (Xj ) . = ψjfi,j−1(Yi,j−1). By the condition L.3, xi ∈ Yi,j−1 and yj = fi,j−1(xi). By condition L.5, it follows that φi(xi) = φjfi,j−1(xi) = φj (yj ). Therefore x = y. • β is surjective. Let z ∈ Z be an arbitrary point. Then z = ψi(xi) for some xi ∈ Xi. Take x = φi(xi) and we have β(x) = z. The uniqueness is trivial. � 3. Fundamental limit space In this section, we define the main concept of this paper, namely, the fun- damental limit space for a closed injective system. Definition 3.1. Let {X,φi} be a limit space for the CIS {Xi,Yi,fi}. We say that X has the weak topology (induced by the collection {φi}i∈N) if the following sentence is true: A ⊂ X is closed in X ⇔ φ−1i (A) is closed in Xi for every i ∈ N. When this occurs, we say that {X,φi} is a fundamental limit space for {Xi,Yi,fi}. Proposition 3.2. Let {X,φi} be fundamental limit space for the CIS {Xi,Yi,fi}. Then φi(Xi) is closed in X for every i ∈ N. Proof. We will prove that φ−1j (φi(Xi)) is closed in Xj for any i,j ∈ N. We have φ−1j (φi(Xi)) =    Xi if i = j ∅ if i < j, fi and fj not semicomponible ∅ if i > j, fj and fi not semicomponible fi,j−1(Yi,j−1) if i < j and fi and fj are semicomponible fj,i−1(Yj,i−1) if i > j and fj and fi are semicomponible In the first three cases it is obvious that φ−1j (φi(Xi)) is closed in Xj . In the fourth case we have the following: If j = i + 1, then fi,j−1(Yi,j−1) = fi(Yi), which is closed in Xi+1, since fi is a closed map. For j > i + 1, since fi is continuous and Yi+1 is closed in Xi+1, then Yi,i+1 = f −1 i (Yi+1) is closed in Xi. Thus, since fi is closed, Lemma 2.2 shows that fi,i+1(Yi,i+1) = fi+1fi(Yi,i) = fi+1fi(Yi), which is closed in Xi+1. Again by Lemma 2.2 we have fi,j−1(Yi,j−1) = fj−1fi,j−2(Yi,j−2). Thus, by induction it follows that fi,j−1(Yi,j−1) is closed in Xj . The fifth case is similar to the fourth. � Corollary 3.3. Let {X,φi} be a fundamental limit space for the CIS {Xi,Yi,fi}. If X is compact, then each Xi is compact. Proof. Each Xi is homeomorphic to the closed subspace φi(Xi) of X. � Proposition 3.4. Let {X,φi} and {Z,ψi} be two limit spaces for the CIS {Xi,Yi,fi}. If {X,φi} is a fundamental limit space for {Xi,Yi,fi}, then the bijection β : X → Z in Theorem 2.9 is continuous. Proof. Let A be a closed subset of Z. We have β−1(A) = ∪∞i=0φi(ψ −1 i (A)) and φ−1j (β −1(A)) = ψ−1j (A). Since ψj is continuous and X has the weak topology, we have that β−1(A) is closed in X. � 100 M. C. Fenille Theorem 3.5 (Uniqueness of the fundamental limit space). Let {X,φi} and {Z,ψi} be two fundamental limit spaces for the CIS {Xi,Yi,fi}. Then, the bijection β : X → Z in Theorem 2.9 is a homeomorphism. Moreover, β is the unique homeomorphism from X onto Z such that ψi = β ◦ φi for every i ∈ N. Proof. Let β′ : Z → X be the inverse map of the bijection β. By preceding proposition, β and β′ are both continuous maps. Therefore β is a homeomor- phism. The uniqueness is the same of Theorem 2.9. � Theorem 3.6 (Existence of fundamental limit space). Every closed injective system has a fundamental limit space. Proof. Let {Xi,Yi,fi} be an arbitrary CIS. Define X̃ = X0∪f0 X1 ∪f1 X2 ∪f2 · · · to be the quotient space obtained of the coproduct (or topological sum) ∐∞ i=0 Xi by identifying each Yi ⊂ Xi with fi(Yi) ⊂ Xi+1. Define each ϕ̃i : Xi → X̃ to be the projection from Xi into the quotient space X̃. Then {X̃,ϕ̃i} is a fundamental limit space for the given CIS {Xi,Yi,fi}. � The latter two theorems implies that every CIS has, up to homeomorphisms, a unique fundamental limit space. This will be remembered and used many times in the article. 4. Examples of CIS’s and limit spaces In this section we show some interesting examples of limit spaces. The first example is very simple and the second shows the existence of a limit space which is not a fundamental limit space. This example will be highlighted in the last section of the article. The other examples show known spaces as fundamental limit spaces. Example 4.1 (Identity limit space). Let {Xi,Yi,fi} be the CIS with Yi = Xi = X and fi = idX for every i ∈ N, where X is an arbitrary topological space and idX : X → X is the identity map of X. It is easy to see that {X,idX} is a fundamental limit space for {Xi,Yi,fi}. Example 4.2 (Existence of limit space which is not a fundamental limit space). Assume X0 = [0, 1) and Y0 = {0}. Take Xi = Yi = [0, 1] for each i ≥ 1. Let f0 : Y0 → X1 be the inclusion f(0) = 0 and fi = identity for each i ≥ 1. Consider the sphere S1 as a subspace of R2. Define the maps φ0 : X0 → S 1 by φ0(t) = (cos πt,− sin πt) and φi : Xi → S 1 by φi(t) = (cos πt, sin πt), for each i ≥ 1. It is easy to see that S1 = ⋃∞ i=0 φi(Xi) and each φi is an imbedding onto its image. Moreover, φi(Xi) ∩φj (Xj ) . = φjfi,j−1(Yi), which implies condition L.3. Therefore, {S1,φi} is a limit space for the CIS {Xi,Yi,fi}. However, this limit space is not a fundamental limit space, since φ0(X0) is not closed in S 1, (or again, since S1 is compact though X0 is not). (See Figure 1 below). CIS’s and its fundamental limit spaces 101 0 X 1 X 2 Xf 0 1 f id 1 S Figure 1. Limit space (not funda- mental) 0 X 1 X 2 X id X 1 y 0 y Figure 2. Fundamental limit space Now, we consider the subspace X = {(x, 0) ∈ R2 : 0 ≤ x ≤ 1} ∪ {(0,y) ∈ R 2 : 0 ≤ y < 1} of R2. Define the maps ψ0 : X0 → X by ψ0(t) = (0, t) and ψi : Xi → X by ψi(t) = (t, 0), for each i ≥ 1. We have X = ⋃∞ i=0 ψi(Xi), where each φi is an imbedding onto its im- age, such that ψi(Xi) is closed in X. Moreover, since ψi(Xi) ∩ ψj (Xj ) . = ψjfi,j−1(Yi), it follows that {X,ψi} is a fundamental limit space for the CIS {Xi,Yi,fi}. (See Figure 2 above). Note that the bijection β : S 1 → X of Theorem 2.9 is not continuous here. Example 4.3 (The infinite-dimensional sphere S∞). For each n ∈ N, we consider the n-dimensional sphere Sn = {(x1, . . . ,xn+1) ∈ R n+1 : x21 + · · · + x 2 n+1 = 1}, and the “equatorial inclusions” fn : S n → Sn+1, defined by fn(x1, . . . ,xn+1) = (x1, . . . ,xn+1, 0). Then {S n,Sn,fn} is a CIS. Its fundamental limit space is {S∞,φn}, where S ∞ is the infinite-dimensional sphere and, for each n ∈ N, the imbedding φn : S n → S∞ is the natural “equatorial inclusion”. Example 4.4 (The infinite-dimensional torus T ∞). For each n ≥ 1, we con- sider the n-dimensional torus T n = ∏n i=1 S 1 and the closed injections fn : T n → T n+1 given by fn(x1, . . . ,xn) = (x1, . . . ,xn, (1, 0)), where each xi ∈ S1. Then {T n,T n,fn} is a CIS, whose fundamental limit space is {T ∞,φn}, where T ∞ = ∏∞ i=1 S 1 is the infinite-dimensional torus and, for each n ∈ N, the imbedding φn : T n → T ∞ is the natural inclusion φn(x1, . . . ,xn) = (x1, . . . ,xn, (1, 0), (1, 0), . . .). Example 4.3 is a particular case of the following one: Example 4.5 (CW-complexes as fundamental limit spaces for its skeletons). Let K be an arbitrary CW-complex. For each n ∈ N, let Kn be the n-skeleton of K and consider the natural inclusions ln : K n → Kn+1 of the n-skeleton into the (n + 1)-skeleton. If the dimension dim(K) of K is finite, then we put Km = K and lm : K m → Km+1 to be the identity map, for every m ≥ dim(K). It is well known that a CW-complex has the weak topology with respect to their skeletons, that is, a subset A ⊂ K is closed in K if and only if A∩Kn is closed 102 M. C. Fenille in Kn for every n. Thus, {Kn,Kn, ln} is a CIS, whose fundamental limit space is {K,φn}, where each φn : K n → K is the natural inclusions of the n-skeleton Kn into K. For details of the CW-complex theory see [2] or [6]. The example below is a consequence of the previous one. Example 4.6 (The infinite-dimensional projective space RP∞). There is al- ways a natural inclusion fn : RP n → RPn+1, which is a closed injective con- tinuous map. (the n-dimensional projective space RPn is the n-skeleton of (n + 1)-dimensional projective space RPn+1). It follows that {RPn, RPn,fn} is a CIS. The fundamental limit space for this CIS is the infinite-dimensional projective space RP∞. For details about infinite-dimensional sphere and projective plane see [2]. 5. The category of CIS’s and cis-morphisms Let X = {Xi,Yi,fi}i and Z = {Zi,Wi,gi}i be two closed injective systems. By a cis-morphism h : X → Z we mean a collection h = {hi : Xi → Zi}i of closed continuous maps checking the following conditions: 1. hi(Yi) ⊂ Wi for every i ∈ N. 2. hi+1 ◦ fi = gi ◦ hi|Yi for every i ∈ N. This latter condition is equivalent to commutativity of the following diagram for each i ∈ N: Yi fi �� hi|Yi // Wi gi �� Xi+1 hi+1 // Zi+1 We say that a cis-morphism h : X → Z is a cis-isomorphism if each map hi : Xi → Zi is a homeomorphism which carries Yi homeomorphicaly onto Wi. For each arbitrary CIS, say X = {Xi,Yi,fi}i, there is an identity cis- morphism 1 : X → X given by 1i : Xi → Xi equal to identity map for each i ∈ N. Moreover, if h : X(1) → X(2) and k : X(2) → X(3) are two cis-morphisms, then it is clear that its natural composition k ◦ h : X(1) → X(3) is a cis-morphism from X(1) into X(3). Also, it is easy to check that associativity of compositions holds whenever possible: if h : X(1) → X(2), k : X(2) → X(3) and r : X(3) → X(4), then r ◦ (k ◦ h) = (r ◦ k) ◦ h. This shows that the closed injective system and the cis-morphisms form a category, which we denote by Cis. (See [3] for details on basic category theory). CIS’s and its fundamental limit spaces 103 Theorem 5.1. All inductive systems on category Cis admit limit. Proof. Let {X(n), h(mn)}m,n be an inductive system of closed injective system and cis-morphisms. Each X(n) is of the form X(n) = {X (n) i ,Y (n) i ,f (n) i }i and each h(mn) : X(m) → X(n) is a cis-morphism and, moreover, h(pq) ◦ h(qr) = h(pr) for every p,q,r ∈ N. For each m ∈ N, we write h(m) to be h(mn) when m = n + 1. For each i ∈ N, we have the inductive system {X (n) i ,h (mn) i }m,n, that is, the injective system of the topological spaces X (1) i ,X (2) i , . . . and all continuous maps h (mn) i : X (m) i → X (n) i , m,n ∈ N, of the collection h (mn). Now, each inductive system {X (n) i ,h (mn) i }m,n can be consider as the closed injective system {X (n) i ,X (n) i ,h (n) i }n. Let {Xi,ξ (n) i }n be a fundamental limit space for {X (n) i ,X (n) i ,h (n) i }n. Xi h (qm) i // X (m) i h (mn) i // ξ (m) i 44iiiiiiiiiiiiiiiiiiiiii X (n) i h (np) i // ξ (n) i == {{{{{{{{ Then, each ξ (n) i : X (n) i → Xi is an imbedding and we have ξ (m) i = φ (n) i ◦h (mn) i for all m < n. Moreover, Xi has a weak topology induced by the collection {ξ (n) i }n. For any m,n ∈ N, with m ≤ n, we have ξ (m) i (Y (m) i ) = ξ (n) i ◦ h (mn) i (Y (m) i ) ⊂ ξ (n) i (Y (n) i ), by condition 1 of the definition of cis-morphism. Moreover, each ξ (n) i (Y (n) i ) is closed in Xi, since each ξ (n) i is an imbedding. For each i ∈ N, we define Yi = ⋃ n∈N ξ (n) i (Y (n) i ). Then, by preceding paragraph, Yi is a union of linked closed sets, that is, Yi is the union of the closed sets of the ascendent chain ξ (1) i (Y (1) i ) ⊂ ξ (2) i (Y (2) i ) ⊂ · · · ⊂ ξ (m) i (Y (m) i ) ⊂ ξ (m+1) i (Y (m+1) i ) ⊂ · · · Now, since {Xi,ξ (n) i }n is a fundamental limit space for {X (n) i ,Y (n) i ,h (n) i }n, for each m ∈ N, we have (ξ (m) i ) −1(Yi) = (ξ (m) i ) −1(∪n∈Nξ (n) i (Y (n) i )) = Y m i which is closed in X (m) i . Therefore, since Xi has the weak topology induced by the collection {ξ (n) i }n, it follows that Yi is closed in Xi. Now, we will build, for each i ∈ N, an injection fi : Yi → Xi+1 making {Xi,Yi,fi}i a closed injective system. For each i ∈ N, we have the diagram shown below. 104 M. C. Fenille Y (n) i f (n) i �� ξ (n) i // ξ (n) i (Y (n) i ) fi �� X (n) i+1 ξ (n) i+1 // Xi+1 For each x ∈ ξ (n) i (Y (n) i ) ⊂ Xi, there is a unique y ∈ Y (n) i such that ξ (n) i (y) = x. Then, we define fi(x) = (ξ (n) i+1 ◦ f (n) i )(y). It is clear that each fi : ξ (n) i (Y (n) i ) → Xi+1 is a closed injective continuous map, since each ξi and f (n) i are closed injective continuous maps. Now, we define fi : Yi → Xi+1 in the following way: For each x ∈ Yi, there is an integer n ∈ N such that x ∈ ξ (n) i (Y n i ). Then, there is a unique y ∈ Y (n) i such that ξ (n) i (y) = x. We define fi(x) = (ξ (n) i+1 ◦ f (n) i )(y). Each fi : Yi → Xi+1 is well defined. In fact: suppose that x belong to ξ (m) i (Y m i ) ∩ ξ (n) i (Y n i ). Suppose, without loss of generality, that m < n. There are unique ym ∈ Y m i and yn ∈ Y n i such that ξ (m) i (ym) = y = φ (n) i (yn). Then, yn = h (mn) i (ym). Thus, ξ (n) i+1◦f (n) i (yn) = ξ (n) i+1◦f (n) i ◦h (mn) i (ym) = ξ (n) i+1◦h (mn) i+1 ◦f (m) i (ym) = ξ (m) i+1 ◦f (m) i (ym). Now, since each fi : Yi → Xi+1 is obtained from a collection of closed injective continuous maps which coincides on closed sets, it follows that each fi is a closed injective continuous map. This proves that {Xi,Yi,fi}i is a closed injective system. Denote it by X. For each n ∈ N, let E(n) : X(n) → X be the collection E(n) = {ξ (n) i : X (n) i → Xi}i. It is clear by the construction that E(n) is a cis-morphism from X(n) into X. Moreover, we have E(m) = h(mn) ◦ E(n). Therefore, {X,E(n)}n is a direct limit for the inductive system {X(n), h(mn)}m,n. � 6. The transition to fundamental limit space as a functor Henceforth, we will write Top to denote the category of the topological spaces and continuous maps. For each CIS X = {Xi,Yi,fi}i, we will denote its fundamental limit space by £(X). The passage to the fundamental limit defines a function £ : Cis −→ Top which associates to each CIS X its fundamental limit space £(X) = {X,φi}. Theorem 6.1. Let h : X → Z be a cis-morphism between closed injective systems and let £(X) = {X,φi}i and £(Z) = {Z,ψi}i be the fundamental limit spaces for X and Z, respectively. Then, there is a unique closed continuous map £h : X → Z such that £h ◦ φi = ψi ◦ hi for every i ∈ N. CIS’s and its fundamental limit spaces 105 Proof. Write h = {hi : Xi → Zi}i. We define the map £h : X → Z as follows: First, consider £(X) = {X,φi} and £(Z) = {Z,ψi}. For each x ∈ X, there is xi ∈ Xi, for some i ∈ N, such that x = φi(xi). Then, we define £h(x) = ψi ◦ hi(xi). This map is well defined. In fact, if x = φi(xi) = φj (xj ), with i < j, then x ∈ φi(Xi) ∩ φj (Xj ) . = φjfi,j−1(Yi,j−1) and xj = fi,j−1(xi). Thus, ψj ◦ hj(xj ) = ψj ◦ hj ◦ fi,j−1(xi) = ψj ◦ gi,j−1 ◦ hi(xi) = ψi ◦ hi(xi). Now, since £h is obtained from a collection of closed continuous maps which coincide on closed sets, £h is a closed continuous map. Moreover, it is easy to see that £h is the unique continuous map from X into Z which verifies, for each i ∈ N, the commutativity £h ◦ φi = ψi ◦ hi. � Sometimes, we write £h : £(X) → £(Z) instead £h : X → Y . This map is called the fundamental map induced by h. Corollary 6.2. The transition to the fundamental limit space is a functor from the category Cis into the category Top. For details on functors see [3]. Corollary 6.3. If h : X → Z is a cis-isomorphism, then the fundamental map £h : £(X) → £(Z) is a homeomorphism. This implies that isomorphic closed injective systems have homeomorphic fundamental limit spaces. 7. Compatibility of limits In this section, given a CIS X = {Xi,Yi,fi} with fundamental limit space {X,φi}, sometimes we write £(X) to denote only the topological space X. This is clear in the context. Theorem 7.1. Let {X(n), h(mn)}m,n be an inductive system on the category Cis and let {X,E(n)}n its direct limit. Then {£(X (n)), £h(mn)}m,n is an in- ductive system on the category Top, which admits £(X) as its directed limit homeomorphic. Proof. By uniqueness of the direct limit, we can assume that {X, Φ(n)}n is the direct limit constructed in the proof of Theorem 5.1. Then, we have E(n) : X(n) → X given by E(n) = {ξ (n) i : X (n) i → Xi}i, where {Xi,ξ (n) i }n is a fundamental limit space for {X (n) i ,X (n) i ,h (n) i }n. By Theorem 6.1, {£(X(n)), £h(mn)}m,n is a inductive system. For each n ∈ N, write X(n) = {X (n) i ,Y (n) i ,f (n) i }i and £(X n) = {X(n),φ (n) i }i. Moreover, write X = {Xi,Yi,fi}i and £(X) = {X,φi}i. The inductive system {£(X(n)), £h(mn)}m,n can be write as {X (n), £h(mn)}m,n. 106 M. C. Fenille We need to show that there is a collection of maps {ϑ(n) : X(n) → X}n such that {X,ϑ(n)}n is a direct limit for the system {X (n), £h(mn)}m,n. For each x ∈ X(n), there is a point xi ∈ X (n) i , for some i ∈ N, such that x = φ (n) i (xi). We define ϑ (n) : X(n) → X by ϑ(n)(x) = φi ◦ ξ (n) i (xi). The map ϑ(n) is well defined. In fact: If x = φ (n) i (xi) = φ (n) j (xj ), with i ≤ j, then we have x ∈ φ (n) i (X (n) i ) ∩ φ (n) j (X (n) j ) . = φ (n) j f (n) i,j−1(Y (n) i,j−1) and, moreover, xj = f (n) i,j−1(xi) and xi ∈ Yi,j ⊂ Xi. Now, in the diagram below, the two triangles and the big square are commutative. In it, we write ξ (n) i | and φ (n) i | to denote the obvious restriction. It follows that φj ◦ ξ (n) j (xj ) = φj ◦ ξ (n) j ◦ fi,j−1(n)(xi) = φj ◦ fi,j−1 ◦ ξ (n) i (xi) = φi ◦ ξ (n) i (xi). This is sufficient to prove that the map ϑ(n) is well defined. Moreover, note that this map makes the diagram below in a commutative diagram. Xi fi,j−1 // φi ##G GG GG GG GG Xj φj{{ww ww ww ww w X X(n) ϑ (n) OO Y (n) i,j ξ (n) i | OO φ (n) i | << zzzzzzzz f (n) i,j−1 // X (n) j φ (n) j bbEEEEEEEE ξ (n) j OO Now, by Theorem 6.1 we have £h(mn) ◦ φ (m) i = φ (n) i ◦ h (n) i for all integers m < n, since £(Xn) = {X(n),φ (n) i }i. Let x ∈ X(m) be an arbitrary point. Then, there is xi ∈ X (m) i such that x = φ (m) i (xi). Also, for each n ∈ N with m < n, we have £h (mn)(x) = φ (n) i ◦ h (mn) i (xi). Thus, we have, ϑ(n) ◦ £h(mn)(x) = φi ◦ ξ (n) i (h (mn) i (xi)) = φi ◦ ξ (m)(xi) = ϑ (m)(x). This shows that ϑ(n) ◦ £h(mn) = ϑ(m) for all integers m < n. Let A be a closed subset of X. Then it is clear that (φi ◦ξ (n) i ) −1(A) is closed in X (n) i , since φi and ξ (n) i are continuous maps. Now, we have (ϑ (n))−1(A) = φ (n) i ((φi ◦ ξ (n) i ) −1(A)). Then, since φ (n) i is an imbedding (and so a closed map), it follows that (ϑ(n))−1(A) is a closed subset of X(n). Therefore, ϑ(n) is continuous. Now, it is not difficult to prove that {X,ϑ(n)}n satisfies the universal map- ping problem (see [3]). This concludes the proof. � CIS’s and its fundamental limit spaces 107 8. Inductive closed injective systems In this section, we will study a particular kind of closed injective systems, which has some interesting properties. More specifically, we study the CIS’s of the form {Xi,Xi,fi}, which are called inductive closed injective system, or an inductive CIS, to shorten. In an inductive CIS {Xi,Xi,fi}, any two injections fi and fj, with i < j, are componible, that is, the composition fi,j = fj ◦ · · · ◦ fi is always defined throughout domain Xi of fi. Hence, fixing i ∈ N, for each j > i we have a closed injection fi,j : Xi → Xj+1. Because this, we define, for each i < j ∈ N, fii = idXi : Xi → Xi and f j i = fi,j−1 : Xi → Xj By this definition, it follows that fki = f k j ◦ f j i , for all i ≤ j ≤ k. Therefore, {Xi,f j i } is an inductive system on the category Top. We will construct a direct limit for this inductive system. Let ∐ Xi = ∐∞ i=0 Xi be the coproduct (or topological sum) of the spaces Xi. Consider the canonical inclusions ϕi : Xi → ∐ Xi. It is obvious that each ϕi is a homeomorphism onto its image. Over the space ∐ Xi consider the relation ∼ defined by: x ∼ y ⇔ { ∃ xi ∈ Xi,yj ∈ Xj with x = ϕi(xi) e y = ϕj (yj ), such that yj = f j i (xi) if i ≤ j and xi = f i j (yj ) if j < i. . Lemma 8.1. The relation ∼ is an equivalence relation over ∐ Xi. Proof. We will check the veracity of the properties reflexive, symmetric and transitive. Reflexive: Let x ∈ X be a point. There is xi ∈ Xi such that x = ψi(xi), for some i ∈ N. We have xi = f i i (xi). Therefore x ∼ x. Symmetric: It is obvious by definition of the relation ∼. Transitive: Assume that x ∼ y and y ∼ z. Suppose that x = ϕi(xi) and y = ϕj (yj ) with yj = f j i (xi). In this case, i ≤ j. (The other case is analogous and is omitted). Since y ∼ z, we can have: Case 1 : y = ϕj (y ′ j ) and z = ϕk(zk) with j ≤ k and zk = f k j (y ′ j ). Then ϕj (yj ) = y = ϕj (y ′ j ), and so yj = y ′ j . Since i ≤ j ≤ k, we have zk = f k j (yj ) = fkj f j i (xi) = f k i (xi). Therefore x ∼ z. Case 2: y = ϕj (y ′ j ) and z = ϕk(zk) with k < j and y ′ j = f j k (zk). Then yj = y ′ j , as before. Now, we have again two possibility: (a) If i ≤ k < j, then f j k (zk) = yj = f j i (xi) = f j k fki (xi). Thus zk = f k i (xi) and x ∼ z. (b) If k < i ≤ j, then f j i (xi) = yj = f j k (zk) = f j i f i k(zk). Thus xi = f i k(zk) and x ∼ z. � Let X̃ = ( ∐ Xi)/ ∼ be the quotient space obtained of ∐ Xi by the equiv- alence relation ∼, and for each i ∈ N, let ϕ̃i : Xi → X̃ be the composition 108 M. C. Fenille ϕ̃i = ρ ◦ ϕi, where ρ : ∐ Xi → X̃ is the quotient projection. ϕ̃i : Xi ϕi // ∐ Xi ρ // X̃ Note that, since X̃ has the quotient topology induced by projection ρ, a subset A ⊂ X̃ is closed in X̃ if and only if ϕ̃i −1 (A) is close in Xi for each i ∈ N. Given x,y ∈ ∐ Xi with x,y ∈ Xi, then x ∼ y ⇔ x = y. Thus, each ϕ̃i is one-to-one fashion onto ϕ̃i(Xi). Moreover, it is obvious that X̃ = ∪ ∞ i=0ϕ̃i(Xi). These observations suffice to conclude the following: Theorem 8.2. {X̃,ϕ̃i} is a fundamental limit space for the inductive CIS {Xi,Xi,fi}. Moreover, {X̃,ϕ̃i} is a direct limit for the inductive system {Xi,f j i }. For details on direct limit see [3]. Remark 8.3. If we consider an arbitrary CIS {Xi,Yi,fi}, then the relation ∼ is again an equivalence relation over the coproduct ∐ Xi. Moreover, in this circumstances, if ϕi(xi) = x ∼ y = ϕj (yj ), then we must have: (a) If i = j, then x = y. (b) If i < j, then fi and fj−1 are semicomponible and xi ∈ Yi,j−1; (c) If i > j, then fj and fi−1 are semicomponible and yj ∈ Yj,i−1. Therefore, it follows that the space X̃ = ( ∐ Xi)/ ∼ is exactly the attaching space X0 ∪f0 X1 ∪f1 X2 ∪f2 · · · , and the maps ϕ̃i are the projections from Xi into X̃, as in Theorem 3.6. 9. Functoriality on fundamental limit spaces Let F : Top → M be a functor of the category Top into a complete category M (a category in which every direct (inductive) or inverse system has a limit). Let {Xi,Xi,fi} be an arbitrary inductive CIS, and consider the inductive system {Xi,f j i } constructed in the previous section. The functor F turns this system into the inductive system {FXi, Ff j i } on the category M. Theorem 9.1 (of the Functorial Invariance). Let {X,φi} be a fundamental limit space for the inductive CIS {Xi,Xi,fi} and let {M,ψi} be a direct limit for {FXi, Ff j i }. Then, there is a unique isomorphism h : FX → M such that ψi = h ◦ Fφi for every i ∈ N. Proof. By Theorem 8.2 and by uniqueness of fundamental limit space, there is a unique homeomorphism β : X → X̃ such that ϕ̃i = β ◦ φi for every i ∈ N. Hence, Fβ : FX → FX̃ is the unique R-isomorphism such that Fϕ̃i = Fβ◦Fφi. Since {X̃,ϕ̃i} is a direct limit for the inductive system {Xi,f j i } on the cate- gory Top, it follows that {FX̃, Fϕi} is a direct limit of the system {FXi, Ff j i } on the category M. By universal property of direct limit, there is a unique isomorphism ω : FX̃ → M such that ψi = ω ◦ Fϕ̃i. Then, we take h : FX → M to be the composition h = ω ◦ Fβ. � CIS’s and its fundamental limit spaces 109 The universal property of direct limits among others properties can be found, for example, in Chapter 2 of [3]. Now, we describe some basic applications of Theorem 9.1. We write Mod to denote the (complete) category of R-modules and R-homomorphisms, where R is a commutative ring with identity element. Example 9.2. Let K be a CW-complex and let {Kn,Kn, ln} be the CIS as in Example 4.5. It is clear that this CIS is an inductive CIS. Let F : Top → Mod be an arbitrary functor. Given m < n in N, write lnm to denote the composition ln−1 ◦ · · · ◦ lm : K m → Kn. Then, {FKn, Flnm} is an inductive system on the category Mod. By Theorem 9.1, its direct limit is isomorphic to FK. Example 9.3 (Homology of the sphere S∞). Let {Sn,Sn,fn} be the CIS of Example 4.3. Its fundamental limit space is the infinite-dimensional sphere S∞. Let p > 0 be an arbitrary integer. By previous example, Hp(S ∞) is isomorphic to direct limit of inductive system {Hp(S n),Hp(f n m)}, where f n m = fn−1 ◦ · · · ◦ fm : S m → Sn, for m ≤ n. Now, since Hp(S n) = 0 for n > p, it follows that Hp(S ∞) = 0 for each p > 0. Details on homology theory can be found in [1], [2] and [5]. Example 9.4 (The infinite projective space RP∞ is a K(Z2, 1) space). We know that π1(RP n) ≈ Z2 for all n ≥ 2 and π1(RP 1) ≈ Z. Moreover, for integers m < n, the natural inclusion fnm : RP m →֒ RPn induces a isomorphism (fnm)# : π1(RP m) ≈ π1(RP n). For details see [2]. The fundamental limit space for the CIS {RPn, RPn,fn} of Example 4.6 is the infinite projective space RP∞. By Example 9.2, we have that π1(RP ∞) is isomorphic to direct limit for the inductive system {π1(RP n), (fnm)#}. Then, by previous arguments it is easy to check that π1(RP ∞) ≈ Z2. On the other hand, for each r > 1, we have πr(RP n) ≈ πr(S n) for every n ∈ N (see [2]). Then, πr(S n) = 0 always that 1 < r < n. Thus, it is easy to check that πr(RP ∞) = 0 for each r > 1. For details on homotopy theory and K(π, 1) spaces see [2] and [6]. Example 9.5 (The homotopy groups of S∞). Since πr(S n) = 0 for all integers r < n, it is very easy to prove that πr (S ∞) = 0 for every r ≥ 1. Example 9.6. The homology of the torus T ∞. Some arguments very simple and similar to above can be used to prove that H0(T ∞) ≈ R and Hp(T ∞) ≈ ⊕∞ i=1 R for every p > 0. 10. Counter-Funtoriality on fundamental limit spaces Let G : Top → M be a counter-functor from the category Top into a com- plete category M (a category in which every direct (inductive) or inverse system has a limit). Let {Xi,Xi,fi} be an arbitrary inductive CIS and consider the inductive system {Xi,f j i } as before. The counter-functor G turns this inductive system on the category Top into the inverse system {GXi, Gf j i } on the category M. 110 M. C. Fenille Theorem 10.1 (of the Counter-Functorial Invariance). Let {X,φi} be a funda- mental limit space for the inductive CIS {Xi,Xi,fi} and let {M,ψi} be an in- verse limit for {GXi, Gf j i }. Then, there is a unique isomorphism h : M → GX such that ψi = Gφi ◦ h for every i ∈ N. Proof. By Theorem 8.2 and by uniqueness of fundamental limit space, there is a unique homeomorphism β : X → X̃ such that ϕ̃i = β ◦ φi, for all i ∈ N. Hence, Gβ : GX̃ → GX is the unique isomorphism such that Gϕ̃i = Gφi◦Gβ. Since {X̃,ϕ̃i} is a direct limit for the inductive system {Xi,f j i } on the category Top, it follows that {GX̃, Gϕi} is an inverse limit for the inverse system {GXi, Gf j i } on the category M. By universal property of inverse limit, there is a unique isomorphism ω : M → GX̃ such that ψi = Gϕ̃i ◦ ω. Then, we take h : M → GX to be the composition h = Gβ ◦ ω. � The property of the inverse limit can be found in [3]. Now, we describe some basic applications of Theorem 10.1. Example 10.2 (Cohomology of the sphere S∞). Since Hp(Sn; R) ≈ Hp(S n; R) for all p,n ∈ Z, it follows by Theorem 10.1 and Example 9.3 that H0(S∞; R) ≈ R and Hp(S∞; R) = 0 for everyp > 0. Example 10.3 (The cohomology of the torus T ∞). Since the homology and cohomology modules of a finite product of spheres are isomorphic, it follows by Theorem 10.1 and Example 9.6 that H0(T ∞) ≈ R and Hp(T ∞) ≈ ⊕∞ i=1 R for every p > 0. 11. Finitely semicomponible and stationary CIS’s We say that a CIS {Xi,Yi,fi} is finitely semicomponible if, for each i ∈ N, there is only a finite number of indices j ∈ N such that fi and fj (or fj and fi) are semicomponible, that is, there is not an infinity sequence {fk}k≥i0 of semicomponible maps. Obviously, {Xi,Yi,fi} is finitely semicomponible if and only if for some (so for all) limit space {X,φi} for {Xi,Yi,fi}, the collection {φi(Xi)}i is a pointwise finite cover of X (that is, each point of X belongs to only a finite number of φi(Xi) ′s). We say that a CIS {Xi,Yi,fi} is stationary if there is a nonnegative integer n0 such that, for all n ≥ n0, we have Yn = Yn0 = Xn0 = Xn and fn = identity map. This section of the text is devoted to the study and characterization of the limit space of these two special types of CIS’s. Theorem 11.1. Let {X,φi} be an arbitrary limit space for the CIS {Xi,Yi,fi}. If the collection {φi(Xi)}i is a locally finite cover of X, then {Xi,Yi,fi} is finitely semicomponible. The reciprocal is true if {X,φi} is a fundamental limit space. Proof. The first part is trivial, since if the collection {φi(Xi)}i is a locally finite cover of X, then it is a pointwise finite cover of X. CIS’s and its fundamental limit spaces 111 Suppose that {X,φi} is a fundamental limit space for the finitely semicom- ponible CIS {Xi,Yi,fi}. Let x ∈ X be an arbitrary point. Then, there are nonnegative integers n0 ≤ n1 such that φ −1 i ({x}) 6= ∅ ⇔ n0 ≤ i ≤ n1. For each n0 ≤ i ≤ n1, write xi to be the single point of Xi such that x = φi(xi). It fol- lows that xi ∈ Yni for n0 ≤ i ≤ n1 − 1, but xn1 /∈ Yn1 and xn0 /∈ fn0−1(Yn0−1). Since fn0−1(Yn0−1) is closed in Xn0 and xn0 /∈ fn0−1(Yn0−1), we can choose an open neighborhood Vn0 of xn0 in Xn0 such that Vn0 ∩ fn0−1(Yn0−1) = ∅. Similarly, since xn1 /∈ Yn1 and Yn1 is closed in Xn1 , we can choose an open neighborhood Vn1 of xn1 in Xn1 such that Vn1 ∩ Yn+1 = ∅. Define V = φn0 (Vn0 ) ∪ φn0+1(Xn0+1) ∪ · · · ∪ φn1−1(Xn1−1) ∪ φn1 (Vn1 ). It is clear that x ∈ V ⊂ X and V ∩ φj (Xj ) = ∅ for all j /∈ {n0, . . . ,n1}. Moreover, we have φ−1j (X − V ) =    Xn0 − Vn0 if j = n0 Xn1 − Vn1 if j = n1 ∅ if n0 < j < n1 Xj otherwise . In all cases, we see that φ−1j (X − V ) is closed in Xj . Thus, X − V is closed in X. Therefore, we obtain an open neighborhood V of x which intersects only a finite number of φi(Xi) ′s. � The reciprocal of the previous proposition is not true, in general, when {X,φi} is not a fundamental limit space. In fact, we have the following example in which the above reciprocal fails. Example 11.2. Consider the topological subspaces X0 = [1, 2] and Xn = [ 1 n+1 , 1 n ], for n ≥ 1, of the real line R, and take Y0 = {1} and Yn = { 1 n+1 } for each n ≥ 1. Define fn : Yn → Xn+1 to be the natural inclusion, for all n ∈ N. It is clear that the CIS {Xn,Yn,fn} is finitely semicomponible, and its fundamental limit space is, up to homeomorphism, the subspace X = (0, 2] of the real line, together the collection of natural inclusions φn : Xn → X. It is also obvious that the collection {φi(Xi)}i is a locally finite cover of X. On the other hand, take Z = ((0, 1] × {0}) ∪ {(1 + cos(πt − π), sin(πt − π)) ∈ R2 : t ∈ [1, 2]}. Consider Z as a subspace of R2. Then Z is homeomorphic to the sphere S1. Consider the maps ψ0 : X0 → Z given by ψ0(t) = (1 +cos(πt−π), sin(πt−π)), and ψn : Xn → Z given by ψn(t) = (t, 0), for each n ≥ 1. It is easy to see that {Z,ψn} is a limit space for the CIS {Xn,Yn,fn}. Now, note that the point (0, 0) ∈ Z has no open neighborhood intercepting only a finite number of ψn(Xn) ′s. Theorem 11.3. Let {X,φi} be a limit space for the CIS {Xi,Yi,fi} and sup- pose that the collection {φi(Xi)}i is a locally finite closed cover of X. Then {X,φi} is a fundamental limit space. Proof. We need to prove that a subset A of X is closed in X if and only if φ−1i (A) is closed in Xi for every i ∈ N. 112 M. C. Fenille If A ⊂ X is closed in X, then it is clear that φ−1i (A) is closed in Xi for each i ∈ N, since each φi is a continuous map. Now, let A be a subset of X such that φ−1i (A) is closed in Xi for every i ∈ N. Then, since each φi is a imbedding, it follows that φi(φ −1 i (A)) = A ∩ φi(Xi) is closed in φi(Xi). But by hypothesis, φi(Xi) is closed in X. Therefore A∩φi(Xi) is closed in X for each i ∈ N. Let x ∈ X − A be an arbitrary point and choose an open neighborhood V of x in X such that V ∩ φi(Xi) 6= ∅ ⇔ i ∈ Λ, where Λ ⊂ N is a finite subset of indices. It follows that V ∩ A = ⋃ i∈Λ V ∩ A ∩ φi(Xi). Now, since each A∩φi(Xi) is closed in X and x /∈ A∩φi(Xi), we can choose, for each i ∈ Λ, an open neighborhood Vi ⊂ V of x, such that Vi∩A∩φi(Xi) = ∅. Take V ′ = ⋂ i∈Λ Vi. Then V ′ is an open neighborhood of x in X and V ′∩A = ∅. Therefore, A is closed in X. � Corollary 11.4. Let {X,φi} be a limit space for the CIS {Xi,Yi,fi} in which each Xi is a compact space. If X is Hausdorff and {φi(Xi)}i is a locally finite cover of X, then {X,φi} is a fundamental limit space. Proof. Each φi(Xi) is a compact subset of the Hausdorff space X. Therefore, each φi(Xi) is closed in X. The result follows from the previous theorem. � Corollary 11.5. Let {X,φi} be a limit space for the finitely semicomponible CIS {Xi,Yi,fi}. Then, {X,φi} is a fundamental limit space if and only if the collection {φi(Xi)}i is a locally finite closed cover of X. Proof. Proposition 3.2 and Theorems 11.1 and 11.3. � Let f : Z → W be a continuous map between topological spaces. We say that f is a perfect map if it is closed, surjective and, for each w ∈ W , the subset f−1(w) ⊂ Z is compact. (See [4]). Let P be a property of topological spaces. We say that P is a perfect property if always that P is true for a space Z and there is a perfect map f : Z → W , we have P true for W . Again, we say that a property P is countable-perfect if P is perfect and always that P is true for a countable collection of spaces {Zn}n, we have P true for the coproduct ∐∞ n=0 Zn. We say that P is finite-perfect if the previous sentence is true for finite collections {Zn} n0 n=0 of topological spaces. Every countable-perfect property is also a finite-perfect property. The reciprocal is not true. Every perfect property is a topological invariant. Example 11.6. The follows one are examples of countable-prefect properties: Hausdorff axiom, regularity, normality, local compactness, second axiom of countability and Lindelöf axiom. The compactness is a finite-perfect property which is not countable-perfect. (For details see [4]). CIS’s and its fundamental limit spaces 113 Theorem 11.7. Let {X,φi} be a fundamental limit space for the finitely semi- componible CIS {Xi,Yi,fi}, in which each Xi has the countable-perfect property P. Then X has P. Proof. Let {X,φi} be a fundamental limit space for {Xi,Yi,fi}. By Theorems 8.2 and 3.5, there is a unique homeomorphism β : X̃ → X such that φi = β◦ϕ̃i for every i ∈ N. Then, simply to prove that X̃ has the property P, where X̃ = ( ∐ Xi)/ ∼ is the quotient space constructed in Section 8 (Remember Remark 8.3). Consider the quotient map ρ : ∐ Xi → X̃. It is continuous and surjective. Moreover, since the CIS {Xi,Yi,fi} is finitely semicomponible, it is obvious that for x ∈ X̃ we have that ρ−1(x) is a finite subset, and so a compact subset, of ∐ Xi. Therefore, simply to prove that ρ is a closed map, since this is enough to conclude that ρ is a perfect map and, therefore, the truth of the theorem. Let E ⊂ ∐ Xi be an arbitrary closed subset of ∐ Xi. We need to prove that ρ(E) is closed in X̃, that is, that ρ−1(ρ(E)) ∩Xi is closed in Xi for each i ∈ N. But note that ρ−1(ρ(E)) ∩ Xi = (E ∩ Xi) ∪ i−1⋃ j=0 fj,i−1(E ∩ Yj,i−1) ∪ ∞⋃ j=i f−1i,j (E ∩ Xj+1), where each term of the total union is closed. Now, since the given CIS is finitely semicomponible, there is on the union ⋃∞ j=i f −1 i,j (E ∩Xj+1) only a finite nonempty terms. Thus, ρ−1(ρ(E)) ∩ Xi can be rewritten as a finite union of closed subsets. Therefore ρ−1(ρ(E)) ∩ Xi is closed. � The quotient map ρ : ∐ Xi → X̃ is not closed, in general. To illustrate this fact, we introduce the following example: Example 11.8. Consider the inductive CIS {Sn,Sn,fn} as in Example 4.3, starting at n = 1. Consider the sequence of real numbers (an)n, where an = 1/n, n ≥ 1. Let A = {an}n≥2 be the set of points of the sequence (an)n starting at n = 2. Then, the image of A by the map γ : [0, 1] → S1 given by γ(t) = (cos t, sin t) is a sequence (bn)n≥2 in S 1 such that the point b = (1, 0) ∈ S1 is not in γ(A) and (bn)n converge to b. It follows that the subset B = γ(A) of S1 is not closed in S1. Now, for each n ≥ 2, let En be the closed (n− 1)-dimensional half-sphere imbedded as the meridian into Sn going by point f1,n−1(bn). It is easy to see that E n is closed in Sn for each n ≥ 2. Let E = ⊔∞ n=2 E n be the disjoint union of the closed half-spheres En. Then, for each n ≥ 2, E ∩ Sn = En and E ∩ S1 = ∅. Thus, E is a closed subset of coproduct space ∐∞ n=1 S n. However, ρ−1(ρ(E)) ∩ S1 = B is not closed in S1. Hence ρ(E) is not closed in the sphere S∞. Therefore, the projection ρ : ∐ Sn → (( ∐ Sn)/ ∼) ∼= S∞ is not a closed map. 114 M. C. Fenille Now, we will prove the result of the previous theorem in the case of sta- tionary CIS’s. In this case the result is stronger and applies to properties finitely perfect. We started with the following preliminary result, whose proof is obvious and therefore will be omitted (left to the reader). Lemma 11.9. Let {X,φi} be a fundamental limit space for the stationary CIS {Xi,Yi,fi}. Suppose that this CIS park in the index n0 ∈ N. Then φi = φn0 for every i ≥ n0 and X ∼= ∪ n0 i=0φi(Xi). Moreover, the composition ρn0 : ∐n0 i=0 Xi inc. // ∐∞ i=0 Xi ρ // X̃ is a continuous surjection, where inc. indicates the natural inclusion. Theorem 11.10. Let {X,φi} be a fundamental limit space for the stationary CIS {Xi,Yi,fi} in which each Xi has the finite-perfect property P. Then X has P. Proof. As in Theorem 11.7, simply to prove that X̃ = ( ∐ Xi)/ ∼ has P. Suppose that the CIS {Xi,Yi,fi} parks in the index n0 ∈ N. By the previous lemma, the map ρn0 : ∐n0 i=0 Xi → X̃ is continuous and surjective. Thus, simply to prove that ρn0 is a perfect map. In order to prove this, it rests only to prove that ρn0 is a closed map and ρ −1 n0 (x) is a compact subset of ∐n0 i=0 Xi, for each x ∈ X̃. This latter fact is trivial, since each subset ρ−1n0 (x) is finite. In order to prove that ρn0 is a closed map, let E be an arbitrary closed subset of ∐n0 i=0 Xi. We need to prove that ρ −1(ρn0 (E)) ∩ Xi is closed in Xi for each i ∈ N. But note that, as before, we have ρ−1(ρn0 (E)) ∩ Xi = (E ∩ Xi) ∪ i−1⋃ j=0 fj,i−1(E ∩ Yj,i−1) ∪ ∞⋃ j=i f−1i,j (E ∩ Xj+1), where each term of this union is closed. Now, since E ⊂ ∐n0 i=0 Xi, we have E ∩ Xj+1 = ∅ for all j ≥ n0. Thus, the subsets f −1 i,j (E ∩ Xj+1) which are in the last part of the union are empty for all j ≥ n0. Hence, ρ −1(ρn0 (E)) ∩ Xi is a finite union of closed subsets. Therefore, ρ−1(ρn0 (E)) ∩ Xi is closed. � References [1] M. J. Greenberg and J. R. Harper, Algebraic Topology, A first course, Ben- jamin/Cummings Publishing Company, London, 1981. [2] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [3] J. J. Hotman, An introduction to homological algebra, Academic Press, Inc., 1979. [4] J. R. Munkres, Topology, Prentice-Hall, 1975. [5] E. H. Spanier, Algebraic Topology, Springer-Verlag New York, Inc. 1966. [6] G. W. Whitehead, Elements of Homotopy Theory, Springer-Verlag New York, Inc. 1978. CIS’s and its fundamental limit spaces 115 Received January 2010 Accepted July 2010 M. C. Fenille (mcfenille@gmail.com) Instituto de Ciências Exatas - Universidade Federal de Itajubá, Av. BPS 1303, Pinheirinho, CEP 37500-903, Itajubá, MG, Brazil. Closed injective systems and its fundamental[8pt] limit spaces. By M. C. Fenille