Spaces with a Finite Family of Basic Functions

Spaces with a Finite F amily of Basic F unctions ∗ P aul Gartside † and Ziqin F eng ‡ Dedicated to Bob Heath on his retiremen t July 20 08 Abstract A T 1 completely regular space X is finite dimensional, locally com- pact and separable metrizable if and only if X h as a fi nite basic family: functions Φ 1 , ..., Φ n such that for all f ∈ C ( X ) there are g 1 , ..., g n ∈ C ( R ) satisfying f ( x ) = P n i =1 g i (Φ i ( x )) for all x ∈ X . This give th e complete solution to four problems on basic functions p osed by S t ernfeld. 1 In tro duction The 13th Pro blem of Hilb ert’s ce le brated list [3] is usua lly interpreted as asking whether ev ery con tin uous real v alued function of three v ariables can b e written as a s uper p o sition (i.e. comp osition) of contin uous functions of tw o v ariables. Kolmogo rov ga ve a strong p ositive solution (w e write C ( X ) for all contin uous real v alued maps on a top olo gical spa ce X , and C ∗ ( X ) for the subset of b ounded maps): Theorem 1 (Kolmogo rov Sup erp osition, [6]) F or a fixe d n ≥ 2 , t her e ar e n (2 n + 1 ) m aps φ pq ∈ C ([0 , 1 ]) su ch that every map f ∈ C ([0 , 1] n ) c an b e written: f ( x ) = 2 n +1 X q =1 g q (Φ q ( x )) wher e Φ q ( x 1 , . . . , x n ) = n X p =1 φ pq ( x p ) , and the g q ∈ C ( R ) ar e maps dep endi ng on f . In addition to solving the sup erp osition problem, K olmogor ov’s theore m says that the functions Φ 1 , . . . , Φ 2 n +1 from [0 , 1] n to the reals form a finite ‘basis’ for all contin uous real v alued maps fro m [0 , 1 ] n . This is a v ery striking phenomena, leading to the follo wing natural definition of Sternfeld [11]: ∗ 2000 Mathematics Su b ject Classificat i on : 26B40, 54C30; 54C35, 54E45. Key W ords and Phrases : Superp osition of functions , finite dimension, lo cally compact, basic family , Hilb ert’s 13th Problem. † Corr e sp onding author Department of Mathematics, University of Pittsburgh, Pittsburgh, P A 15260, USA, email : gartside@math.pitt.edu. ‡ Departmen t of Mathematics, Universit y of Pittsburgh, P A 15260, USA 1 Definition 2 L et X b e a top olo gic al sp ac e. A family Φ ⊆ C ( X ) is said to b e basic (r esp e ctively, basic ∗ ) for X if e ach f ∈ C ( X ) (r esp e ct ively, C ∗ ( X ) ) c an b e written: f = n X q =1 ( g q ◦ Φ q ) , for some Φ 1 , . . . , Φ n in Φ and ‘c o-or dinate functions’ g 1 , . . . , g n ∈ C ( R ) . Beyond their intrinsic interest, ba sic functions hav e prov ed to b e widely us e- ful. Since the use of ba sic functions reduces calculatio ns o f functions simply to addition and ev aluation of a fixed finite family of functions, applica tions to nu merica l ana lysis, approximation and function rec onstruction are immediately apparent. But o ther applications have e merged including to neural net works [4 , 5 , 7]. Extending the Kolmo g orov Superp os itio n Theorem, Ostrand [9] show ed tha t every compact metric space of dimension n has a bas ic family of size 2 n + 1. Subsequently Sternfeld [11] showed that this basic family is minimal in the sense that a compact metric spa c e with a basic family of size no mo re tha n 2 n + 1 must hav e dimension ≤ n . Noting that Dos s [1] had shown that Euclidean n -space, R n , has a basic family of size 4 n for n ≥ 2, Sternfeld asked (in P r oblems 9– 13 o f [11]) ex a ctly which spa c es have a finite basic family , and whether the minimal size of a basic family on a space X was 2 n + 1 where n = dim( X ). Hattori [2] show ed that ev ery lo cally compact, sepa rable metrizable space X of dimension n has a finite bas ic ∗ family of size 2 n + 1. He asked whether the re striction to lo cally compact s paces w as necessary . Our Main Theorem below gives a stro ng and complete solution to all these problems. Since spaces with a finite basic family are finite dimensional, it seems plau- sible that spaces with a countable basic fa mily would b e countable dimens ional. But w e pr ov e that if a space has a coun table basic f amily , then so me finit e sub c ollection is a lso basic, and s o the space is finite dimensional (and lo ca lly compact, sepa rable metrizable). T o facilitate the pro of, and provide full gen- erality we m ake the following definitio n allowing mo re general superp os ition representations than a ‘basic’ repre s ent ation. Definition 3 L et X b e a top olo gic al sp ac e. A family Φ ⊆ C ( X ) is said to b e generating (r esp e ctively, generating ∗ ) for X with r esp e ct to a ‘set of op- er ations’ M of c ontinuous functions mapping fr om su bsets of Euclide an sp ac e into su bsets of Euclide an sp ac e, if e ach f ∈ C ( X ) (r esp e ctively, C ∗ ( X ) ) c an b e written as a c omp osition of functions fr om Φ , M and C ( R ) . Note that a basic (respec tively , basic ∗ ) family is g enerating (resp ectively , g enerating ∗ ) with resp ect to M = { + } , and clearly ‘generating’ implies ‘generating ∗ ’. Theorem 4 (Main Theorem) L et X b e T 1 and c ompletely r e gular. Then the fol lowing ar e e quivalent: 1) X has a c ountable gener ating ∗ family with r esp e ct to a c ountable set of op er ations, 2) X h as a finite b asic family, and 3) X i s a finite dimensional, lo c al ly c omp act and sep ar able metrizable. 2 F urther, a lo c al ly c omp act sep ar able metrizable sp ac e X has dimension ≤ n if and only if it has a b asic family of size ≤ 2 n + 1 . By the preceding note, 2) = ⇒ 1) is immediate. In the next section (Section 2) we prov e 1 ) = ⇒ 3), in Section 3 we establish 3) = ⇒ 2), and we justify the ‘F ur ther’ claim characterizing dimensio n in Section 4. 2 Restrictions Induced b y G enerating ∗ F ami lies In this section, all topolo gical spaces are T 1 and completely regular. Lemma 5 L et X have a gener ating ∗ family Φ with r esp e ct to M . Then e : X → R Φ define d by e ( x ) = (Φ( x )) Φ ∈ Φ is an emb e dding. Pro of. Clearly e is co nt inuous (each pro jection is a Φ in Φ which is co n- tin uous). It is also ea sy to s ee e is injective. T ake distinct x, x ′ in X . Pick f ∈ C ∗ ( X ) such that f ( x ) = 0, f ( x ′ ) = 1 . Represen t f as a comp o sition of Φ 1 , . . . , Φ n in Φ , member s of M and C ( R ). If e ( x ) = e ( x ′ ) then Φ i ( x ) = Φ i ( x ′ ) for all i , and so f ( x ) = f ( x ′ ), whic h is a contradiction. It remains to s how that the top olog y induced on X by e contains the o riginal top ology . Since X is completely regular it is sufficient to c heck that for every f ∈ C ∗ ( X ) the ma p e ( f ) : e ( X ) → R defined by e ( f )( x ) = f ( e − 1 ( x )) is contin uous. But eac h f ∈ C ∗ ( X ) can be written as a comp osition of some Φ 1 , . . . , Φ n in Φ and members of M and C ( R ). Note that for each i we have Φ( e − 1 ( x )) = π Φ i ( x ), where π Φ i is the pro jection map of R Φ onto the Φ i th co - ordinate. Hence e ( f ) = f ◦ e − 1 is the comp ositio n of con tinuous maps, namely the π Φ i s and functions in M and C ( R ), and so is contin uous as required. Since any subspace o f R N is separa ble metrizable, a nd any subspace of R n is finite dimensional w e deduce from Lemma 5: Corollary 6 a) A sp ac e with a c oun table gener ating ∗ family is sep ar able met rizable. b) A sp ac e with a fin ite gener ating ∗ family is fin ite dimensional. A subspace C of a space X is sa id to b e C ∗ -embedded in X if every f ∈ C ∗ ( C ) can be extended to a contin uous b ounded real v alued function on X . In a normal spa ce all closed s ubs paces are C ∗ -embedded. Compact subspa c e s are alwa ys C ∗ -embedded. W e note the following easy lemma: Lemma 7 If Φ is a gener ating ∗ family (r esp e ctively, b asic ∗ ) f or a sp ac e X with r esp e ct to M , and C is C ∗ -emb e dde d in X t hen Φ | C = { Φ | C : Φ ∈ Φ } is a gener ating ∗ (r esp e ctively, b asic ∗ ) family for C . Lemma 8 A sp ac e with a c ountable gener ating ∗ family is lo c al ly c omp act. 3 Pro of. Suppose the space X has a countable generating ∗ family Φ with resp ect to M , but is not lo cally compact. Since X is metrizable, it follo ws that the metric fa n F (defined b elow) embeds a s a clo sed subspace in X . Hence by Lemma 7 it suffices to show that F does not admit a countable g e nerating ∗ family (with resp ect to any set of op eratio ns M ). The metric fan F ha s underlying s et {∗} ∪ ( N × N ) and top ology in which all p oints other than ∗ are isolated and ∗ has bas ic neighborho ods B ( ∗ , N ) = {∗} ∪ ([ N , ∞ ) × N ). F or a co nt ra dic tio n, let Φ = { Φ 1 , Φ 2 , . . . } be a countable generating ∗ family with resp ect to M . F o r each i , let y i = Φ i ( x 0 ), and pick basic op en U i containing ∗ s uch tha t Φ i ( U i ) ⊆ ( y 1 − 1 , y 1 + 1). Now for each n let V n = T n i =1 U i . So Φ i ( V n ) ⊆ ( y i − 1 , y i + 1) for i = 1 , . . . , n . W e can write V n = { ∗} ∪ ([ N n , ∞ ) × N ) and suppo se, without loss of generality , that N n > N m if n > m . Fix n . Let D 0 = { x 0 k = ( N n , k ) : k ∈ N } . As { Φ 1 ( x 0 k ) } k ∈ N is a subset of [ y 1 − 1 , y 1 + 1], which is sequentially compac t, there is a D 1 = { x 1 k : k ∈ N } ⊆ D 0 such that { Φ 1 ( x 1 k ) } k ∈ N is co nv ergent. As { Φ 2 ( x 1 k ) } k ∈ N is a subset of [ y 2 − 1 , y 2 + 1], whic h is sequen tially compact, there is a D 2 = { x 2 n : n ∈ N } ⊆ D 0 such that { Φ 2 ( x 2 k ) } k ∈ N is conv erg ent. Inductively we get D n = { x n k : k ∈ N } , which is infinite c losed d iscr ete and for each i = 1 , ..., n the sequence { Φ i ( x n k ) } k ∈ N is conv ergent, say to z n i . Define D n O = { x n 2 k − 1 : k ∈ N } and D n E = { x n 2 k : k ∈ N } . Define f : F → [0 , 1] by: f is iden tically zero outside S n D n O (in particular , f is zero o n each D n E ), and f is identically 1 /n on D n O . Then f is co nt inuous and bounded. Hence, for so me m , f can b e written as the comp osition o f Φ 1 , . . . , Φ m and mem b ers of M a nd C ( R ). Now, on the o ne hand lim k Φ i ( x m 2 k − 1 ) = z i.m = lim k Φ i ( x m 2 k ) so b y contin uit y of the elements of M and C ( R ) in the comp osi- tional representation of f , lim k f ( x m 2 k − 1 ) = lim k f ( x m 2 k ), and on the other hand, lim k f ( x m 2 k − 1 ) = 1 /m 6 = 0 = lim k f ( x m 2 k ). This is our desired contradiction. Let Y b e a lo ca lly c o mpact separable metrizable space. W rite C k ( Y ) for C ( Y ) with the compact-op en topolo gy . Then C k ( Y ) is a P olish (separable, completely metrizable) gro up. In par ticula r, for any n , C k ( R ) n is a Polish group. Lemma 9 If X has a c ountable gener ating ∗ family with r esp e ct to a c ountable set of op er ations, M , then X has a finite gener ating ∗ family with r esp e ct to a finite set of op er ations M ′ . Pro of. Let Φ 1 , Φ 2 , . . . b e a countable generating ∗ family fo r X with resp ect to the countable set of op erations M . By Lemma 8 X is lo ca lly compact and C k ( X ) is a Polish group. Let g 1 , g 2 , . . . be formal letters representing functions fro m R to R . Let W be the set of all for ma l comp ositions of Φ i s, elements of M and g i s. Note that W is countable. Fix w in W . Then w induces a map ( g 1 , . . . , g n ) 7→ w ( g 1 , . . . , g n ) from C k ( R ) n → C k ( X ) where we substitute actual g i ∈ C ( R ) fo r the cor resp onding 4 formal letter. This map is contin uous with resp ect to the compact-op en top ol- ogy . Let F w = w ( C k ( R ) n ). It is analy tic. Define G w = F w ∩ C k ( X, (0 , 1 )). Since C k ( X, (0 , 1 )) is homeomo rphic to C k ( X ) it is Polish, and hence must b e a G δ subset of C k ( X ). So G w is analytic in C k ( X, (0 , 1 )). Note, b y the generating ∗ prop erty , that C ∗ k ( X ) ⊆ S w ∈ W F w . Hence C k ( X, (0 , 1 )) = S w ∈ W G w . By the Baire C a tegory theo rem there m ust be some particular w in W such that G w is not meager. Fix a homeomor phism h : R → (0 , 1 ). Via h , addition and subtraction on R induce (co ntin uous) group op erations ⊕ , ⊖ : (0 , 1 ) × (0 , 1) → (0 , 1). These op erations on (0 , 1 ) in turn induce o p e rations on C k ( X, (0 , 1 )) making this space a Polish gro up. Let H w be the subgroup of C k ( X, (0 , 1 )) genera ted b y G w . By Pettis’ Theo- rem [10], since G w is non-meag er a nd analytic, G w ⊖ G w has non-empty interior. Hence the subgro up H w is op en, and so coinc ides with C k ( X, (0 , 1 )) (whic h is connected). Set Φ ′ to be the finite set of Φ i s app ear ing in w , and set M ′ to b e ⊕ , ⊖ and the finite s et of elemen ts of M app earing in w . Since H w = C ( X , (0 , 1)), each element of C ( X , (0 , 1)) is a comp osition of members of Φ ′ , M ′ and C ( R ). W e check Φ ′ is a finite ge ne r ating ∗ family with resp ect to M ′ . F or if f ∈ C ∗ ( X ), then f ma ps into some open interv al ( a, b ). Fix a homeomorphism g 0 : R → R taking (0 , 1) to ( a, b ). Then f = g 0 ◦  g − 1 0 ◦ f  , where g − 1 0 ◦ f is in C k ( X, (0 , 1 )). Hence g − 1 0 ◦ f can b e expr essed a s a co mp os ition of elements o f Φ ′ , M ′ and so me g 1 , . . . , g n in C ( R ). But now f is g 0 of this comp osition and so is also expressible in terms of elemen ts of Φ ′ , M ′ and C ( R ), as required. W e no te that the finite genera ting ∗ family is a s ubset of the or ig inal family , and also that if the or iginal family is generating then we can take M ′ ⊆ M ∪ { + , −} . Pro of o f 1) = ⇒ 3). Let X be a spa c e with a c ountable genera ting ∗ family with res pe c t to a count- able set of ope r ations. By Co rollar y 6 a ) X is s eparable metriz a ble. Lemma 8 then says that X is lo cally compact. F rom Lemma 9 we deduce that X ha s a finite generating ∗ family . Hence by Corollary 6 b) X is finite dimensional. 3 Construction of Finite Basic F amilie s This section is dev oted to proving: Lemma 10 If X is a lo c al ly c omp act, sep ar able metrizable sp ac e of dimensiosn ≤ n then X h as a b asic family of size 2 n + 1 . Lemma 10 is the forward implication o f the ‘F ur ther’ claim in the Main Theor em. The implication ‘3) = ⇒ 2)’ of the Main Theore m then follows. Recall that Hattori [2] show ed that ev ery lo cally co mpact, sepa rable metriz- able space X of dimension n has a finite basic ∗ family of size 2 n + 1. Lemma 10 5 and its pro of impr ov es on Hattori’s r e sult a nd pro of b ecause: (1) it applies to all functions (not necessar ily b ounded), (2) it is cons tructive (Hattori’s argu- men t uses a Baire category argument) and (3 ) it is cons iderably shorter tha n Hattori’s. The pro o f is similar to that of Ostrand for compact metr ic s paces. How ever difficulties ar is e be cause contin uous rea l v alued functions on a lo cally compact space need not be b ounded . F o r this sec tion, fix a lo cally co mpact, separable spac e X of dimension ≤ n , and with compatible metric d . W e can find { K m : m ≥ − 1 } a countable cov er o f X by compact sets such that K − 1 = K 0 = ∅ and K m ⊆ K 0 m +1 for each m ≥ − 1 . F o r eac h m ≥ 0 w e put H m = K m \ K ◦ m − 1 , and set U m = K ◦ m +1 \ K m − 1 . Since Ostrand has done the compac t case, w e c an as s ume that the K m ’s ar e strictly increasing. W e show X ha s a basic family of size 2 n + 1. The bas ic functions Φ i are defined to be the limit of approximations f i k . The approximations a r e defined inductiv ely along with some families o f ‘nice’ cov ers. These ‘nice’ cov ers co me from Ostrand’s [9] c haracter ization o f dimension: Theorem 11 (Os trand’s Cov ering Theorem) A metric sp ac e Y of dimen- sion ≤ n if and only if for e ach op en c over C of Y and e ach inte ger k ≥ n + 1 ther e ex ist k discr ete fa milies of op en sets U 1 , . . . , U k such that the u nion of any n + 1 of the U i is a c over of Y which r efines C . Lemma 12 L et γ > 0 . Ther e ar e 2 n + 1 many families S 1 , . . . , S 2 n +1 of op en subsets of X , and η m > 0 for m ≥ 0 , satisfying: (1) Each S i is discr ete in X. (2) F or k fixe d and e ach x ∈ X fixe d, |{ S ∈ S 2 n +1 i =1 S i : x ∈ S }| ≥ n + 1 . (3) diam S < γ for any S ∈ S 2 n +1 i =1 S i . (4) S 2 n +1 i =1 S i r efines { U m : m ∈ ω } . (5) F or any m ∈ N , { S : S ∈ S 2 n +1 i =1 S i , S ∩ K m 6 = ∅ } is finite. (6) S ( H m , η m ) ∩ S = ∅ if H m ∩ S = ∅ for any S ∈ S n +1 i =1 S i . (7) S ( H m − 1 , η m − 1 ) ∩ S ( H m +1 , η m +1 ) = ∅ . In (6) and ( 7), S ( H m , η m ) = { x ∈ X : d ( H m , x ) ≤ η m } Pro of. Let C = { C a : a ∈ N } b e a locally finit e o p en co ver of X with: diam ( C a ) < γ and |{ H m : H m ∩ C a 6 = ∅}| ≤ 2, for each a ∈ N . Then b y Ostrand’s covering theorem, there ex ist 2 n + 1 discrete fa milies of open se ts S 1 , · · · , S 2 n +1 which r efines C . Also the union of any n + 1 of the S i is a cover of X . So 1 )-4) are easy to v erify . Fix i with 1 ≤ i ≤ 2 n + 1. As S i is discr ete, { S : S ∩ K m 6 = ∅ , S ∈ S i } is finite. Thus condition 5) is satisfied. Now fix i and m , the discreteness of S i guarantees th at H m ∩ [ { S : S ∈ S i and H m ∩ S = ∅} = ∅ . So d ( H m , S { S : S ∈ S i and H m ∩ ( S ) = ∅} > 0 . Then w e can pick η m i such that S ( H m , η m i ) ∩ S = ∅ if H m ∩ S = ∅ for any S ∈ S i . Let η m = min { η m i : i = 1 , · · · , 2 n + 1 } . This sa tisfies 6). 6 Notice that since H m is co mpact for ea ch m ∈ N , w e can pic k η m small enough such that S ( H m − 1 , η m − 1 ) ∩ S ( H m +1 , η m +1 ) = ∅ , giving (7). Step 1: Construction of the appro ximations Again, we generalize the construction of Ostrand, but must find wa ys around the pr oblem of not having bo unded functions. By induction o n k ≥ 0 , using Lemma 12, for i = 1 , ..., 2 n + 1 , ther e exis t: po sitive real n umbers ǫ k with ǫ 1 < 1 / 4, γ k , η m k distinct positive prime n um b ers r i k , discrete families S 1 k , ..., S 2 n +1 k and contin uous functions f i k : X → [0 , k + 1], with the following prop erties . F o r each k ∈ N , the families S 1 k , ..., S 2 n +1 k , γ k and η m k satisfy (1)–(7) of Lemma 12. F urther: (A) lim k →∞ γ k = lim k →∞ ǫ k = 0; (B) ǫ k < 1 / Π 2 n +1 i =1 r i k ; (C) f i k is constant on the closure of tho se mem b ers of S i k which hav e nonempt y int erse c tion with K m for ( m ≤ k ), the constant b eing a n in tegr al multiple of 1 /r i k , and takes differen t v alues on distinct mem b ers. Then we can take a con- tin uous extension of f i k to the rest of the space. (D) F or any S in S i k having nonempty in tersectio n with H m , m − 1 < f i m ( S ) < m + 1 . Also for m ≥ 2, by (7), w e can make m − 1 < f i k ( S ( H m , η m k ) < m + 1. F o r each i ∈ N , if S ∩ H m 6 = ∅ and S ∩ H m +1 6 = ∅ , then m < f i m ( C ) < m + 1 ;if S ∩ H m 6 = ∅ and S ∩ H m − 1 6 = ∅ , then m − 1 < f m ( S ) i < m ; (E) F o r each ℓ < j < k and x ∈ K ℓ , f i j ( x ) < f i k ( x ) < f i j ( x ) + ǫ j − ǫ k for any i . Step 2: Co nstruction of the basic function s F rom (E), for any x ∈ K m and k > m , f i m ( x ) < f i k ( x ) < f i m ( x ) + ǫ 1 for an y i = 1 , . . . , 2 n + 1. Thus we can take the uniform limit of f i k restricted o n K m . F or any x ∈ K m let Φ i ( x ) = lim k →∞ f i k ( x ). So Φ i is contin uous on K m for ea ch m . Hence Φ i is contin uous on X . Also by (D) fo r x ∈ H m , m − 1 < Φ i ( x ) < m + 1 + 1 / 4. Let V i k = { Φ i ( S ) : S ∈ S i k } . Then if S ∩ K m 6 = ∅ a nd S ∈ S i k with k > m , Φ i ( S ) is contained in the interv a l [ f i k ( S ) , f i k ( S ) + ǫ k ] by (E). By (B), these closed interv als a re disjoint for ea ch fixed m and k with k ≥ m . Then each V i k is discrete. Step 3: Cons truction of the co ordinate functions T ake any function f ∈ C ( X ). W e find g 1 , . . . , g 2 n +1 ∈ C ( R ) such that f = P 2 n +1 i =1 g i ◦ Φ i . F o r each s ≥ 0, de fine the compact subset L s = K s +1 \ K ◦ s − 1 . Since K 1 is compact and K 1 ⊆ K ◦ 2 , there exists a function f 1 such that f 1 ( x ) = f ( x ) for x ∈ K 1 and f 1 ( x ) = 0 for x ∈ X \ K ◦ 2 . Then letting g 1 = f − f 1 , it is easy to see tha t g 1 ( x ) = 0 for x ∈ K 1 . Similarly , ther e exists f 2 such that f 2 ( x ) = g 1 ( x ) for x ∈ K 2 and f 2 ( x ) = 0 for x ∈ X \ K ◦ 3 . Inductively , f can be written as a n infinite sum P ∞ s =1 f s such that f s ( x ) = 0 for x ∈ X \ L s . F o r ea ch s , f s is b ounded and uniformly co ntin uous. Fix s ∈ N . Note that for each x ∈ L s , s − 2 < Φ i ( x ) < s + 2 + 1 / 4. 7 By construction, if we restrict the discrete families S 1 , · · · , S 2 n +1 and the functions Φ 1 , · · · , Φ 2 n +1 to K s +1 , then the discr ete families and functions are exactly those defined b y Ostrand [9]. In par ticular, the functions Φ 1 | L s , . . . , Φ 2 n +1 | L s are basic fo r L s (Lemma 2). Thu s we can re present f s | L s ( x ) = P 2 n +1 i =1 g s i (Φ i | L s ( x )), for so me g s i ∈ C ( R ). W e can redefine g s i to b e constantly zero outside o f [ s − 2 , s + 2 + 1 / 4] because the image of Φ i is contained in [ s − 2 , s + 2 + 1 / 4] and f s ( x ) = 0 if x ∈ L s \ ( L s ) ◦ . Now f s = P 2 n +1 i =1 g s i ◦ Φ i . Finally , letting g i = P ∞ s =1 g s i , we see that g i is con tin uous because g i ( x ) is a finite sum of non-zero contin uous functions for each x ∈ R , a nd f = P 2 n +1 i =1 g i ◦ Φ i – as required. 4 Characterization of Dimension Lemma 10 says that a lo cally co mpact, separa ble metrizable spa ce o f dimension ≤ n has a basic family of size ≤ 2 n + 1, giving the forward implica tion in the ‘F ur ther’ of Theorem 4. F or the conv erse: Lemma 13 A sp ac e X with a b asic ∗ family Φ 1 , . . . , Φ N , wher e N ≤ 2 n + 1 , has dimension ≤ n . Pro of. T ake any compact subset K of X . By Lemma 7, the maps Φ 1 | K, . . . , Φ N | K form a bas ic ∗ family for K , hence by compactness a basic family . By Sternfeld’s result connecting dimensio n and basic families in compact spaces , it follows that dim K ≤ n . By Lemma 8, X is lo ca lly compac t, separ able metrizable. Hence it has a lo cally fin ite cov er by compa c t se ts – eac h, by the ab ov e, o f dimensio n ≤ n . By the Lo cally Finite Sum Theorem for dimensio n, we deduce that X itself m ust hav e dim ensio n ≤ n . References [1] R. Dos s, A sup erp osition theorem for unbounded contin uous functions. T rans. Amer. Math. Soc. 2 3 3 (1977), 197–20 3 [2] Y. Hattor i, Dimension and super po sition of bo unded contin uous functions on lo cally compact, separable metric space s. T op olo gy Appl. 54 (1993 ), no. 1-3, 123– 132 [3] D. H ilb ert, Mathematische Probleme, Nac hr. Ak ad. Wies. Gottingen (1900), 25 3 29 7, Ges ammelte Abhandlungen, Bd. 3 , Springer, Ber lin, 1935, pp. 290-3 2 9. [4] R. Hec ht-Nielsen. 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