An Inverse System of Nonempty Objects with Empty Limit

AN INVER SE SYSTEM OF NONEMPT Y OBJE CTS WITH EMPTY LIMIT SA TY A DEO AND V EERENDRA VIKRAM A W ASTHI Abstract. In this a rticle we give an explicit example of an inverse system with nonempty sets and onto bo nding maps suc h that its inv erse limit is empty . 1. Introduction It has b een often quoted, without g iving an example, that there are in v erse systems in whic h all the ob jects are no nempty and the b onding maps are on t o, but the in vers e limit of the in ve rse system is empt y . (see e.g.[3], Dugundji [1] pg 4 2 7, la st para). It is hard to b elieve that there should b e an example like this, but the fact is that t here exists suc h an example. The original pap er by L. Henkin [2] dealing with this problem giv es a theorem whic h implies that there are sev era l examples, but the explanation and pro of of the theorem is to o abstract to ha v e a clear idea of a sp ecific example. In this note, w e presen t a concrete example of a n in v erse system of sets and maps ha ving the stated prop erty , whic h can b e easily under- sto o d. It will then easily follo w that w e can ha ve an e xample of suc h an in v erse system in any category admitting arbitrary pro ducts, e.g., in the category of mo dules and homomorphisms, category of top olog ical spaces and contin uous maps, etc. 2. Not a tions and Preliminaries First, let us ha ve the following w ell kno wn definitions (see [1]): Definition 2.1. A binary relation R in a set A is called a preorder if it is reflexiv e and transitiv e. A set together with a definite preorder is called a preordered set . 2000 Mathematics S ubje ct Classific ation. 18G0 5,18B0 5 and 16B5 0 . Key wor ds and phr ases. Inv ers e sy stem, Inv er se limit, Ordina l num b ers. 1 2 SA TY A DEO AND VEEREND RA VIKR AM A W ASTHI Definition 2.2. Let A b e a preor dered set and { X α | α ∈ A} b e a family of to p ological spaces indexed b y A . F or each pair of indices α, β satisfying α < β , a ssume that there is a giv en a con tin uous maps f β α : X β → X α and that these maps satisfy the following condition: If α < β < γ , then f γ α = f β α of γ β . Then the family { X α ; f β α } is called an inv erse system ov er A w ith top ological spaces X α and bonding con tinuous maps f β α . Definition 2.3. Let X = { X α, f β α ; α ≤ β } α,β ∈A b e an inv erse system of t o p ological spaces and contin uous map f β α : X β → X α , α ≤ β based on index ing set A . Consider the pro duct space Q α ∈A X α , a nd let p α : Π X α → X α denotes the pro jection map. Define X ′ = { x ∈ Y α ∈A X α | whenev er α ≤ β , f β α p β ( x ) = p α ( x ) } . Then the set X ′ with subspace top ology is called the in v erse limit of the inv erse system X . Remark 2.1. W e ha ve de fined ab o v e an in vers e system and in vers e limit in the category of top olog ical spaces. Clearly , this definition of an inv erse system can b e made in a n y category . Ho wev er, the in v erse limit of an in vers e system will exist only in those categories whic h admit arbitrary pro ducts. Ordinal Numbers : F or definition and well kno wn sp ecial pro p erties of ordinal num b ers w e refer to D ugundji [1]. The successor x + of a s et x is defined a s x ∪ { x } , and then ω w as constructed as the smallest set that contains 0 and that con tains x + whenev er it con tains x . Now , the question arises that what happ ens if w e start with ω , form its successor ω + , then form the successor of t ha t, a nd pro ceed so on. In ot her words, is there something out b ey ond ω , ω + , ( ω + ) + , · · · , etc., in the same sense in whic h ω is b ey ond 0 , 1 , 2 , · · · , etc.? W e mention the names of some of the first few of them. After 0 , 1 , 2 , · · · comes ω , and after ω , ω + 1 , ω + 2 , · · · comes 2 ω . After 2 ω + 1 (that is, the success o r of 2 ω ) comes 2 ω + 2 , and then 2 ω + 3 ; nex t after all the terms of the sequ ence so b egun comes 3 ω . A t this p oin t another application of axiom of substitution is required. Next comes 3 ω + 1 , 3 ω + 2 , 3 ω + 3 , · · · , and after them comes 4 ω . In this w ay w e get successiv ely ω , 2 ω , 3 ω , 4 ω , · · · . . An applicatio n of axiom of substi- tution yields something that follow s them all in the same sense in whic h ω follows the na t ural n um b ers: tha t something is ω 2 . After the whole AN IN VERSE SYS TEM OF N ONEMPTY OBJECTS WITH EMPTY LI MIT 3 thing starts o v er again : ω 2 + 1 , ω 2 + 2 , ω 2 + 3 , ω 2 + ω , ω 2 + ω + 1 , ω 2 + ω + 2 , · · · , ω 2 + 2 ω , ω 2 + 2 ω + 1 , ω 2 + 2 ω + 2 , · · · , ω 2 + 3 ω , · · · , ω 2 + 4 ω , 2 ω 2 , · · · , 3 ω 2 , · · · , ω 3 , · · · , ω 4 , · · · , ω ω , · · · , ω ( ω ω ) , · · · . Since the countable union of coun table sets is again countable, eac h of the ab o v e num b ers is countable. Therefore, using the w ell-ordered prop ert y of ordinals there exists a smallest ordinal n umber ω 1 whic h con tains all of the ab ov e n um b ers and is itself uncountable. W e call ω 1 as the first uncoun table or dina l n um b er. 3. The Example Let { 0 , 1 , 2 , · · · , ω , ω +1 , · · · , 2 ω , 2 ω + 1 , · · · , ω 2 , ω 2 + 1 , · · · } b e the set of ordinal n um b ers a nd ω 1 b e the first uncoun table ordinal. Consider the set Ω = [0 , ω 1 ) of a ll ordinals less than ω 1 . W e will cons truct an in ve rse mapping system { X α , f β α ; α ≤ β } based on the directed set Ω in whic h all the sets X α will b e nonempty , the b onding maps f β α : X β → X α , α ≤ β , will b e on to, but lim ← − X α = φ . Let us define a p oint to mean a finite sequence of an eve n n um b er of elemen ts fro m Ω , e.g., x = ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n ) , whic h satisfy t he follow ing three conditions: (i) α 1 < α 2 (ii) α 2 i − 1 < α 2 i +2 for 0 < i < n, i.e., α 1 < α 4 , α 3 < α 6 , α 5 < α 8 , · · · (iii) α 2 i +1 < α 2 i +2 and α 2 i +1 ≮ α 2 j +1 for 0 ≤ j < i < n, where α ≮ β holds when neither α < β nor α = β . This means α 1 < α 2 , α 3 < α 4 , α 5 < α 6 , α 7 < α 8 , · · · a nd α 3 ≮ α 1 , α 5 ≮ α 3 , α 5 ≮ α 1 , · · · . W e ma y observ e that the ab ov e conditions imply that α 1 < α 2 , α 1 < α 4 , α 3 < α 4 , α 3 < α 6 , α 5 < α 6 , α 5 < α 8 , α 7 < α 8 , α 7 < α 10 , and so on. W e define index of the p oin t x giv en ab ov e to b e α 2 n − 1 , order of x to b e α 2 n and length of x t o b e n . Let us illustrate a few b egining sets X α , α ∈ Ω (i) x ∈ X 0 , means x = (0 , α ) where α > 0 . 4 SA TY A DEO AND VEEREND RA VIKR AM A W ASTHI (ii) x ∈ X 1 , means t he p oint x can b e one of t he follow ing types x = (1 , α ) where α > 1 , or x = (0 , α , 1 , β ) where α > 0 and β > 1. (iii) x ∈ X 2 , means t he p oint x can b e one of t he follow ing types x = (2 , α ) where α > 2 , or x = (0 , α , 2 , β ) where α > 0 and β > 2 o r x = (1 , α , 2 , β ) where α > 1 and β > 2 o r x = (0 , α , 1 , β , 2 , γ ) where α > 0 a nd β > 1 and γ > 2 Note that all the p oin ts defined ab o v e hav e index 2. Similar ly w e can c hec k the elemen ts of X 3 as follows: (iv) x ∈ X 3 means x is one of the fo llo wing t yp es x = (3 , α ) where α > 3 or x = (0 , α , 3 , β ) where α > 0 and β > 3 o r x = (1 , α , 3 , β ) where α > 1 and β > 3 o r x = (2 , α , 3 , β ) where α > 2 and β > 3 or x = (0 , α , 1 , β , 3 , γ ) where α > 0 a nd β > 1 and γ > 3 o r x = (0 , α , 2 , β , 3 , γ ) where α > 0 a nd β > 2 and γ > 3 o r x = (1 , α , 2 , β , 3 , γ ) where α > 1 a nd β > 2 and γ > 3 o r x = (0 , α, 1 , β , 2 , γ , 3 , δ ) where α > 0 and β > 1 and γ > 2 and δ > 3 , α, β , γ , δ ∈ Ω . Th us w e ha ve a family of nonempt y disjoint sets X α , α ∈ Ω , whose elemen ts are the p oin ts with index α. W e now define the b onding maps f β α : X β → X α , α ≤ β , α, β ∈ Ω . Let x = ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n ) b e an a rbitrary p oint in X β (so that α 2 n − 1 = β ). W e define the image of x in X α under f β α as follows: There are tw o cases : Case I: If α ≤ α 1 , then we define f β α ( x ) = ( α , α 2 ) a nd since x is a p oin t in X β , α 1 < α 2 b y condition (i) which implies α < α 2 . Therefore ( α, α 2 ) is a p oint with index α . Hence ( α , α 2 ) ∈ X α . Case I I: If α ≮ α 1 then there exist a least j, 0 < j < 2 n − 1 suc h t hat α < α 2 j +1 b ecause α < β = α 2 n − 1 . Then w e define f β α ( x ) = f β α ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n ) = ( α 1 , α 2 , · · · , α 2 j , α, α 2 j +2 ) . Clearly , ( α 1 , α 2 , · · · , α 2 j , α, α 2 j +2 ) satisfies all the three conditions of a p oin t as it is o nly a subsequence of the p o in t x . Also ( α 1 , α 2 , · · · , α 2 j , α, α 2 j +2 ) has index α hence ( α 1 , α 2 , · · · , α 2 j , α, α 2 j +2 ) ∈ X α . Note that in particular, the map f 3 2 : X 3 → X 2 , will b e as follows. AN IN VERSE SYS TEM OF N ONEMPTY OBJECTS WITH EMPTY LI MIT 5 (3 , α ) 7→ (2 , α ) ∈ X 2 (Case I) (0 , α , 3 , β ) 7→ (0 , α, 2 , β ) ∈ X 2 (1 , α , 3 , β ) 7→ (1 , α, 2 , β ) ∈ X 2 (2 , α , 3 , β ) 7→ (2 , α ) ∈ X 2 (Case I) (0 , α , 1 , β , 3 , γ ) 7→ (0 , α , 1 , β , 2 , γ ) ∈ X 2 (0 , α , 2 , β , 3 , γ ) 7→ (0 , α , 2 , β ) ∈ X 2 (1 , α , 2 , β , 3 , γ ) 7→ (1 , α , 2 , β ) ∈ X 2 (0 , α , 1 , β , 2 , γ , 3 , δ ) 7→ (0 , α , 1 , β , 2 , γ ) ∈ X 2 . All others are as in Case I I. Ha ving defined f β α , α ≤ β , let us note the following ob vious prop erties of these b onding maps: (i). f α α : X α → X α is iden tity . (ii). Let f β α : X β → X α and f γ β : X γ → X β , α < β < γ , α , β , γ ∈ Ω b e tw o b onding maps. Then f β α of γ β = f γ α : X γ → X α . Let us ve rify a sp ecific example of (ii): Let f 6 4 : X 6 → X 4 and f 4 3 : X 4 → X 3 b e the t wo b onding maps. W e wan t to verify f 4 3 f 6 4 = f 6 3 . F or this we choose an ar bitr ary elemen t of X 6 sa y , x = (0 , α , 1 , β , 3 , γ , 5 , δ, 6 , ω ) ∈ X 6 , where α > 0 , β > 1 , γ > 3 , δ > 5 , and ω > 6 . Then b y the definition of f β α w e hav e f 6 4 ( x ) = (0 , α, 1 , β , 3 , γ , 5 , δ ) = y , sa y a nd f 4 3 ( y ) = (0 , α, 1 , β , 3 , γ ) . Also, f 6 3 ( x ) = f 6 3 (0 , α , 1 , β , 3 , γ , 5 , δ, 6 , ω )) = (0 , α , 1 , β , 3 , γ ) . Hence f 4 3 f 6 4 = f 6 3 . Th us w e hav e the follow ing inv erse sys tem { X α , f β α ; α ≤ β , α , β ∈ Ω } of sets and maps defined on the directed set Ω . · · · X γ f γ β − → X β f β α − → X α · · · X 2 f 2 1 − → X 1 , where 1 < 2 < · · · < α < β < γ and 1 , 2 , · · · , α, β , γ ∈ Ω No w we verify tha t the b onding maps f β α : X β → X α , α ≤ β , are onto: 6 SA TY A DEO AND VEEREND RA VIKR AM A W ASTHI Let x = ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n ) ∈ X α so that α = α 2 n − 1 . W e c ho ose γ > β and then consider the sequence of ev en n umber of ele men ts from the directed set Ω and let y = ( α 1 , α 2 , · · · , α 2 n , β , γ ). Sin ce x is a subseque nce of y and x is a p oin t and a lso α 2 n − 1 < α < β < γ , to pro ve t ha t y is a p oint in X β , it suffices to v erify that β ≮ α 2 j +1 , 0 ≤ j < n . But if this is true that β ≤ α 2 j +1 , 0 ≤ j < n will imply that α < α 2 j +1 since α < β , whic h is contradiction to the fact that x = ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n ) is a p oint. Thu s y = ( α 1 , α 2 , · · · , α 2 n , β , γ ) is a p oint a nd it is an ele ment of X β suc h that f β α ( y ) = x b y the definition of f β α . Let us see a part icular example of on toness as fo llows : Consider f 5 3 : X 5 → X 3 . W e will sho w that there is preimage of all the elemen ts of X 3 (all the 8 ty p es a s discuss ed earlier), is in X 5 . The preimage of any elemen t of X 3 , x = ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n ) where α 2 n − 1 = 3 can b e obta ined by just intro ducing tw o more elemen ts from the directed set Ω in p oint x as ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n , 5 , δ ) , δ > 5 . Th us w e see ( f 5 3 ) − 1 ( x ) = ( f 5 3 ) − 1 ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n ) = ( α 1 , α 2 , · · · , α 2 n − 1 , α 2 n , 5 , δ ) = y ∈ X 5 So, w e hav e now a n in v erse mapping system based on Ω { X α , f β α : X β → X α , α ≤ β , α, β ∈ Ω } , in whic h each X α 6 = φ and each f β α : X β → X α , α ≤ β is on t o . 4. Proof of the main re sul t W e claim that for the ab ov e in ve r se sys tem X , lim ← − X α = φ . Let us a ssume the con trary and supp ose there exists an elemen t x in the inv erse limit lim ← − X α . In other words, x ∈ Q X α where x = ( x 0 , x 1 , · · · , x ω , x ω +1 , · · · , x 2 ω , x 2 ω +1 , · · · ) , x i ∈ X i ( ∗ ) suc h that whenev er α ≤ β , f β α ( x β ) = x α . Let Ø b e the set of o rders of these x ′ i s in ( ∗ ). It is clear that this set Ø is a cofinal set o f Ω, since for any α ∈ Ω there is a set X α and since x ∈ Q X α , the i-th co-ordinate in x is some elemen t x α ∈ X α . Since α =index of x α < order of x α ∈ Ø, w e find that Ø is a cofinal set in Ω . AN IN VERSE SYS TEM OF N ONEMPTY OBJECTS WITH EMPTY LI MIT 7 W e also observ e that if length of x α = length of x β , then order of x α = order of x β . T o pro v e this we c ho ose a γ > α and γ > β . Then there exists an elemen t x γ ∈ X γ in ( ∗ ) such that f γ α ( x γ ) = x α and f γ β ( x γ ) = x β . But from the definition of the b onding map f β α it follo ws that the orders o f x α and x β are some elemen t in the sequence x γ , sa y order of x α = α 2 i and order of x β = α 2 j . Th us if the length of x α = i = length o f x β = j, then clearly , α 2 i = α 2 j i.e., order of x α = order o f x β . Therefore, the set of orders Ø b ehav es according to the lengths of x i in ( ∗ ), and length of any p oin t is a natural num b er. Th us, if the lengths of x ′ i s are un b ounded then we will get a simple sequence o f the o rders of the elemen ts x i whic h will b e cofinal. Hence there will exist a cofinal simple sequence in Ω = [0 , ω 1 ). But this is clearly a contradiction. On the other hand if the lengths of x ′ i s are b ounded, then the set of orders of x ′ i s, i.e., Ø will contain a maximal elemen t of Ω = [0 , ω 1 ) whic h is ag a in a con tradiction. Hence, we find that in either of the tw o cases viz., when the lengths are un b o unded or b ounded w e ha v e a con tradiction since it is w ell kno wn tha t the set Ω = [0 , ω 1 ) neither p osseses a simple cofinal sequence nor a maximal elemen t. H ence there can not exist an y elemen t in the inv erse limit, i.e., lim ← − X α = φ . Remark 4.1. In view of the ab ov e construction, it is clear that one can alw ays hav e a n in vers e system in an y cat ego ry (e.g., top ological spaces and con tinuous functions or mo dules and mo dule homomorphisms etc.) admitting arbitrary pr o ducts with nonempt y ob jects and onto b o nding morphism whose in v erse limit can b e empt y . Reference s [1] JAMES DUGUNDJI, T op olo gy . McGraw-Hill, (1966) [2] LEON HENKIN, A pr oblem on inv erse mapping sy stems. Pro c. Amer. Math. So c. 1 , (1950). 2 24–22 5. [3] N. E . STEE NR O D AND S. EILENBE R G, F oundations of Algebraic T opo logy . Princeton Univ Pr ess, (1952). Sat ya Deo, Harish Chandra Researc h Institute, Chhatnag Road, Jh usi, Allahabad 211 019, India. Email: sdeo@mri.ernet.in, v csdeo@y aho o.com V eerendra Vikram Aw a sthi, Institute of Mathematical Sciences, 8 SA TY A DEO AND VEEREND RA VIKR AM A W ASTHI CIT Campus, T aramani Chennai 600 0 1 3, India. Email: vv aw asthi@imsc.res .in, vv a wasthi@y aho o.com