EXTENSION DIMENSION OF INVERSE LIMITS. University of Zagreb, Croatia

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1 GLASNIK MATEMATIČKI Vol. 35(55)(2000), EXTENSION DIMENSION OF INVERSE LIMITS Sbe Mardešć Unversty of Zagreb, Croata Abstract. Recently L.R. Rubn and P.J. Schapro have consdered nverse sequences X of metrzable spaces X, whose extenson dmenson dm X P,.e., P AE(X ), where P s an arbtrary polyhedron (or CW -complex). They proved that dm X P, where X = lm X. The present paper generalzes ther result to nverse sequences of stratfable spaces, gvng at the same tme a more conceptual proof. 1. Introducton By a polyhedron P we mean the geometrc realzaton K of a smplcal complex K endowed wth the CW- topology. We say that the extenson dmenson of a space X does not exceed P, and we wrte dm X P, provded every mappng f : A P from a closed subset A X to P admts an extenson to all of X,.e., P s an absolute extensor for X, P AE(X). Formally, extenson dmenson (for compacta) was frst ntroduced n a 1994 paper by A. Dranshnkov [9]. It was further studed by A. Dranshnkov and J. Dydak [10] and other authors. A classcal theorem of dmenson theory asserts that, for normal spaces X, the coverng dmenson dm X n f and only f dm X S n (see e.g., Theorem of [12]). If G s an abelan group and K = K(G, n) s an Elenberg - MacLane complex, then for paracompact spaces X, the cohomologcal dmenson dm G X n f and only f dm X K. Ths follows from the work of H. Cohen [7], P.J. Huber [13], E.G. Sklyarenko [19] and Y. Kodama [14] Mathematcs Subject Classfcaton. 54B35, 54C55, 54F45. Key words and phrases. nverse lmt, coverng dmenson, cohomologcal dmenson, extenson theory, extenson dmenson, metrzable space, stratfable space. 339

2 340 SIBE MARDEŠIĆ It s well known that for nverse systems X = (X λ, p λλ, Λ) of compact Hausdorff spaces wth dm X λ n the nverse lmt X has dmenson dm X n (see e.g., Theorem of [12]). The followng general proposton s also easly proved (see Theorem 2.2 of [8]). Proposton 1. Let X = (X λ, p λλ, Λ) be an nverse system of compact Hausdorff spaces wth nverse lmt X and let P be a polyhedron. If dm X λ P, for every λ Λ, then also dm X P. Much deeper s a 1959 result of K. Nagam [16] (also see Theorem of [12]), whch asserts that the lmt X of an nverse sequence X = (X, p +1 ) of metrzable spaces X wth dmenson dm X n has dmenson dm X n. Its generalzaton to an arbtrary polyhedron P and nverse sequences of metrzable spaces such that dm X P, for every N, was recently proved by L.R. Rubn and P.J. Schapro [18]. The case of separable metrc spaces was obtaned earler by A. Chgogdze [6]. In ths theorem (as well as n Nagam s theorem) the assumpton that the spaces X are metrzable cannot be replaced by the weaker condton that the spaces X be paracompact. Indeed, n 1980 M.G. Charalambous exhbted an nverse sequence of paracompact 0-dmensonal spaces X, whose lmt X s a normal space and dm X > 0 [5]. The purpose of the present paper s to generalze the Rubn Schapro theorem to nverse sequences of stratfable spaces and, more mportant, to gve a more conceptual proof. The followng s our man result. Theorem 1. Let P be a polyhedron and let X = (X, p +1 ) be an nverse sequence of stratfable spaces wth lmt X. If dm X P, for all, then also dm X P. The proof of Theorem 1 s a modfcaton of the natural proof of Proposton 1. Therefore, we frst outlne that proof. Proof of Proposton 1. Let A X be a closed set and let f : A P be a mappng. Choose an open coverng V of P such that any two V-near mappngs nto P are homotopc. Consder the nverse system A = (A λ, p λλ, Λ), where A λ = p λ (A) (p λ : X X λ are the natural projectons) and p λλ : A λ A λ are the restrctons to A λ of p λλ : X λ X λ. It s readly seen that A s the lmt of A wth projectons p λ : A A λ, whch are the restrctons to A of the projectons p λ : X X λ. Snce A s compact and A s an nverse system of compact spaces, p = (p λ ): A A s a resoluton (see Theorem 1 of I.6.1 n [15]). Therefore, by a characterzaton of resolutons (see I.6.2 of [15]), there exst a λ Λ and a mappng g λ : A λ P such that the mappng g = g λ p λ : A P s V-near to f. Clearly, g factors through A λ and, by the choce of V, t s homotopc to f. Snce dm X λ P, g λ extends to a mappng h λ : X λ P. Therefore, h λ p λ : X P s an extenson of g to all of X. Now

3 EXTENSION DIMENSION OF INVERSE LIMITS 341 the homotopy extenson theorem mples that also f admts an extenson to all of X. Remark 1. Note that whenever a mappng g : A P factors through some A λ, then t also factors through all A λ, where λ ranges through a cofnal subset of Λ. Indeed, t suffces to consder ndces λ λ and put g λ = g λ p λλ. In the case of an nverse system X = (X λ, p λλ, Λ) of non-compact spaces wth lmt X and projectons p λ : X X λ we must replace factorzaton of mappngs by the more general noton of a fltered factorzaton of mappngs, defned as follows. Let X = (X λ, p λλ, Λ) be an nverse system of spaces wth lmt X and projectons p λ : X X λ. Let U X be a non -empty open set and let g : U P be a mappng. By a fltered factorzaton of g through X we mean a famly of open sets G λ X λ, λ Λ, and a famly of mappngs g λ : G λ P whch satsfy the followng condtons. (1.1) λλ (G λ) G λ, λ < λ, (1.2) λ (G λ) = U, λ Λ (1.3) g λ (G λ) = g λ p λ λ (G λ). Condton (1.1) mples (1.4) λ (G λ) λ (G λ ), λ λ, whch together wth (1.2) shows that the sets λ (G λ) form an ncreasng fltraton of U. On the other hand, (1.3) gves factorzatons of g restrcted to members of that fltraton. Some of the sets G λ can be empty, but not all because of (1.2). Remark 2. If the projectons p λ are surjectve, condton (1.3) mples (1.5) g λ λλ (G λ ) = g λ p λλ λλ (G λ ), λ λ. The key step n the proof of Theorem 1 s the followng theorem whch could prove useful n other stuatons as well. Theorem 2. Let X = (X, p +1 ) be an nverse sequence of paracompact perfectly normal spaces wth lmt X and surjectve projectons p : X X. Let P be a polyhedron, V an open coverng of P and U X an open set. Then every mappng f : U P admts a mappng g : U P, whch s V-near to f and admts a fltered factorzaton through X. In [17] Rubn has establshed a more general verson of hs result wth Schapro by replacng the condton dm X P, N, by the weaker condton dm X P. For an nverse system X = (X λ, p λλ, Λ) the latter condton

4 342 SIBE MARDEŠIĆ means that, for every λ Λ, closed set A λ X λ and mappng f λ : A λ P, there exsts a λ λ such that the mappng f λ p λλ λλ (A λ) extends to all of X λ. The analogous result for nverse sequences of stratfable spaces s establshed n Theorem 4. The proof also uses Theorem Prelmnares We now recall some well-known notons and facts from geometrc and general topology needed n our proofs. If U s an open coverng of a space X, let N(U) denote the correspondng nerve. A mappng f : X N(U ) s called canoncal f f 1 (St (U, N(U))) U, for every U U. An open coverng s normal f t admts a canoncal mappng. In a paracompact space every open coverng s normal. Every polyhedron P admts an open coverng V such that any two V-near mappngs nto P are homotopc (see e.g., Theorem 2.6 of [3]). Every open coverng V of a polyhedron P admts a trangulaton K such that the closed stars of K refne V (see e.g., Theorem 4, Appendx 1 of [15]). Two mappngs f, g nto the geometrc realzaton K of a smplcal complex K are sad to be contguous, denoted by f g, provded every pont x X admts a smplex σ K such that f(x), g(x) σ, where σ denotes the closure of σ n K. Every paracompact space s normal. Paracompact perfectly normal spaces (open sets are F σ -sets) are heredtarly paracompact,.e., all of ther subsets are paracompact (see e.g., Exercse n [1]). A T 1 -space X s stratfable provded wth every open set U X one can assocate a sequence of open sets U n X n such a way that the followng condtons be fulflled. (S1) U n U, (S2) n=1 U n = U, (S3) U V U n V n. Stratfable spaces were ntroduced n 1961 by J. Ceder [4] as a generalzaton of metrzable spaces. Every stratfable space s paracompact and perfectly normal. Every subset of a stratfable space s stratfable. Hence, stratfable spaces are heredtarly paracompact. The drect sum of an arbtrary collecton of stratfable spaces s stratfable. The drect product of a countable collecton of stratfable spaces s stratfable. Consequently, the lmt of an nverse sequence of stratfable spaces s a stratfable space. All polyhedra are stratfable spaces. Polyhedra are absolute neghborhood extensors (ANE s) for stratfable spaces [3]. It easly follows that polyhedra have the homotopy extenson property for stratfable spaces,.e., f P s a polyhedron, X s a stratfable space and A X s a closed set, then every mappng f : (X 0) (A I) P extends to all of X I.

5 EXTENSION DIMENSION OF INVERSE LIMITS Fltered factorzatons of mappngs Ths secton s devoted to the proof of Theorem 2. For ths we need some lemmas. Lemma 1. Let X, X be topologcal spaces and let p: X X be a surjectve mappng. Let U = (U γ, γ Γ) be a collecton of non-empty open sets n X and let U X be ther unon. Smlarly, let U = (U γ, γ Γ ) be a collecton of non-empty open sets n X and let U X be ther unon. If Γ Γ and (3.6) (U γ) U γ, for γ Γ, then the ncluson Γ Γ nduces a smplcal njecton s: N(U ) N(U) between the correspondng nerves. Furthermore, f f : U N(U) and f : U N(U ) are canoncal mappngs, then f (U ) and sf p (U ) are contguous mappngs nto N(U). Proof. Let the vertces U γ 0,..., U γ n, γ 0,..., γ n Γ, span a smplex of N(U ). Then U γ 0... U γ n. Snce p: X X s a surjecton, t follows that (3.7) (U γ 0... U γ n ) = (U γ 0 )... (U γ n ) U γ0... U γn and thus, the vertces U γ0 = s(u γ 0 ),..., U γn = s(u γ n ) span a smplex of N(U). We wll now prove that the mappngs f (U ) and sf p (U ) are contguous. Let x (U ) and let U γ0,..., U γn be the vertces of the smplex σ N(U) whch contans f(x) n ts nteror. Smlarly, let U γ..., U 0, γ be m the vertces of the smplex σ N(U ) whch contans f p(x) n ts nteror. Snce s s a smplcal njecton, the pont sf p(x) les n the nteror of the smplex σ N(U), whose vertces are U γ 0,..., U γ m. It thus suffces to show that the vertces U γ0,..., U γn, U γ 0,..., U γ m span a smplex of N(U),.e., that (3.8) U γ0... U γn U γ 0... U γ m. Note that (3.9) f(x) St (U γ0, N(U))... St (U γn, N(U)). Snce f s a canoncal mappng, f 1 (St (U γ, N(U)) U γ and thus, (3.9) mples (3.10) x U γ0... U γn. Smlarly, (3.11) f p(x) St (U γ 0, N(U ))... St (U γ m, N(U )) and (3.6) mples (3.12) x (U γ 0 )... p 1 (U γ m ) U γ 0... U γ m.

6 344 SIBE MARDEŠIĆ Clearly, (3.10) and (3.12) yeld (3.8). Lemma 2. Let X = (X, p +1 ) be an nverse sequence of heredtarly paracompact spaces wth lmt X and surjectve projectons p : X X, N. Let U X be an open set and let f : U P be a mappng nto a polyhedron P = K. Then there exst open sets U X such that (3.13) +1 (U ) U +1, (3.14) N (U ) = U. Moreover, there exst maps h : U K, N, such that (3.15) h p (U ) h (U ), (3.16) h p (U ) f (U ). Proof. Let Γ be the set of all vertces of K for whch (3.17) U γ = f 1 (St (γ, K)). Consder the open coverng U = (U γ, γ Γ) of U and note that Γ s a subset of the set K 0 of all the vertces of K. For N and γ Γ consder the open subset U γ of X, defned as the unon of all open sets V X, for whch (V ) U γ. Let Γ Γ be the set of all γ Γ, for whch U γ. Clearly, Γ Γ and (3.18) (U γ ) U γ U, for γ Γ. Put U = (U γ, γ Γ ) and (3.19) U = γ Γ U γ. To prove (3.13) note that p = p +1 p +1 mples (3.20) +1 (p 1 +1 (U γ)) = (U γ ) U γ. Snce U +1γ contans all open subsets of V X +1 satsfyng +1 (V ) U γ, formula (3.20) shows that (3.21) +1 (U γ) U +1γ. Note that (3.21) mples Γ Γ +1. Indeed, f γ Γ,.e., U γ, then +1 (U γ), because the surjectvty of p mples the surjectvty of p +1. Consequently, U +1γ,.e., γ Γ +1. Fnally, (3.21), (3.19) for and + 1 and the ncluson Γ Γ +1 yeld (3.13). To establsh (3.14) frst note that (3.18) and (3.19) mply (U ) U. To prove the converse ncluson consder a pont x U and choose γ Γ so that x U γ. Snce X = lm X and U γ s open, there s an N and there

7 EXTENSION DIMENSION OF INVERSE LIMITS 345 s an open set V X such that p (x) V and (V ) U γ. Therefore, V U γ and γ Γ. Moreover, (3.22) x (V ) (U γ ) (U ). Let us now see that N(U) can be vewed as a subcomplex of K and f can be vewed as a canoncal mappng f : U N(U) K. To verfy the frst asserton t suffces to see that the ncluson Γ K 0 nduces a smplcal njecton s: N(U) K. Indeed, f U γ0,..., U γn span a smplex of N(U), then U γ0... U γn and thus, (3.23) f 1 (St (γ 0, K)... St (γ n, K)), whch mples St (γ 0, K)... St (γ n, K). Consequently, the vertces γ 0 = s(u γ0 ),..., γ n = s(u γn ) span a smplex of K. Also note that f(u) N(U) K. Indeed, f x U and the vertces γ 0,..., γ n span a smplex σ K whch contans f(x) n ts nteror, then f(x) les n St (γ 0, K)... St (γ n, K) and therefore, x U γ0... U γn. Consequently, U γ0... U γn. However, ths mples that γ 0,..., γ n Γ and U γ0,..., U γn are vertces whch span a smplex of N(U). Moreover, for γ Γ, (3.17) mples (3.24) f 1 (St (U γ, N(U)) f 1 (St (γ, K)) = U γ, whch shows that f : U N(U) s a canoncal mappng. In order to defne the mappngs h : U K we frst choose, for every N, a canoncal mappng g : U N(U ). Such mappngs exst because X s heredtarly paracompact and thus, U s paracompact. We then apply Lemma 1 to the spaces X, X, the mappng p, the open coverngs U of U and U of U and the canoncal mappngs f : U N(U) and g : U N(U ). We obtan a smplcal njecton s : N(U ) N(U) nduced by the ncluson Γ Γ. Defne the desred mappng h : U N(U) P by (3.25) h = s g. One obtans (3.16) as an mmedate consequence of the asserton of Lemma 1. To prove (3.15), apply Lemma 1 to the surjecton p +1, to the collectons U +1 and U and to the canoncal mappngs g +1 : U +1 N(U +1 ) and g : U N(U ). One obtans a smplcal njecton s +1 : N(U ) N(U +1 ), nduced by the ncluson Γ Γ +1, such that (3.26) s +1 g p (U ) g (U ). Also note that s +1 s +1 : N(U ) N(U) s a smplcal mappng nduced by the nclusons Γ Γ +1 Γ. Consequently, t concdes wth s : N(U ) N(U). Note that the compostons of two contguous mappngs wth the same smplcal mappng are contguous mappngs. Therefore, (3.25) and (3.26)

8 346 SIBE MARDEŠIĆ yeld (3.27) h p (U ) = s +1 s +1 g p (U ) s +1 g (U ) = h +1 p+1 1 (U ). Remark 3. Note that the set U γ was defned as the maxmal open set V havng the property that (V ) s contaned n U γ. Rubn and Schapro [18] defned such an open set as the U γ - response to p and denoted t by resp(u γ, p ). We use ths constructon also n the proof of Theorem 3, where V = resp(v, p ). Lemma 3. Let X = (X, p +1 ) be an nverse sequence of perfectly normal spaces wth lmt X and projectons p : X X. Let U X and U X be open sets whch satsfy (3.13) and (3.14). Then there exst open sets G X such that (3.28) G U, (3.29) +1 (G ) G +1, (3.30) (G ) = U. N Proof. In a perfectly normal space Y each open set V s a cozero-set,.e., t s of the form φ 1 (0, 1], for some mappng φ : Y [0, 1] (see e.g., Corollary of [11]). It s therefore easy to see that, for every N, the open set U can be represented as the unon of a sequence of open subsets V 1, V 2,... of X such that (3.31) V n V n V n+1, (3.32) U = j N V j. We defne the open sets G X by nducton on. In addton to condtons (3.28) and (3.29) we also requre that (3.33) k (V k ) G k, for 1 k. Snce V1 1 U 1 and X 1 s normal, t s possble to choose an open set G 1 n X 1 for whch G 1 U 1 and V1 1 G 1. Now assume that we have already defned sets G 1,..., G n accordance wth (3.28), (3.29) and (3.33). We choose as G +1 an open set n X +1 such that G +1 U +1 and G +1 contans the closed sets +1 (G ) and +1 j+1 (Vj ), where 1 j +1. Snce ths s a fnte collecton of closed sets contaned n U +1, the exstence of G +1 s a consequence of the normalty of X +1. Clearly, the sets G constructed n ths way satsfy (3.28), (3.29) and (3.33). It remans to prove that they also satsfy (3.30).

9 EXTENSION DIMENSION OF INVERSE LIMITS 347 Gven a pont x U, (3.14) shows that there exsts an ndex N such that x (U ). Therefore, by (3.32), there exsts an nteger j 1 such that p (x) V j and thus, p (x) V k, for k j. If also k, one has p (x) = p k p k (x) and thus, (3.34) p k (x) k (p (x)) k (V k ). By (3.33), p k (x) G k and thus, x s contaned n k (G k) (G ). Consequently, U s contaned n the left sde of (3.30). The opposte ncluson s an mmedate consequence of G U and of (3.14). Remark 4. The natural analogues of Lemmas 2 and 3 hold also for nverse systems ndexed by cofnte drected sets Λ. Lemma 4. Let X be a normal space and K a smplcal complex. Let A X be a closed set and let V, U X be open sets such that A V V U. If h: U K and g : V K are mappngs such that h V and g are contguous mappngs, then there exsts a mappng k : U K, whch s contguous to h and s such that (3.35) k A = g A, (3.36) k U\V = h U\V. Proof. By normalty of X choose an open set H X such that A H H V. Choose a mappng φ: X [0, 1] such that φ(a) = 1 and φ(x\h) = 0. Then defne k : U K by { φ(x)g(x) + (1 φ(x))h(x), x V, (3.37) k(x) = h(x), x U\H. Note that, for every pont x V, the ponts g(x) and h(x) belong to a closed smplex σ from K. Therefore, φ(x)g(x) + (1 φ(x))h(x) s a well-defned pont of σ. Moreover, the two expressons n (3.37) assume the same values on V (U\H), whch shows that k s a well-defned mappng. Fnally, for x A, φ(x) = 1 and thus, k(x) = g(x). Smlarly, for x U\V, φ(x) = 0 and thus, k(x) = h(x). Proof of Theorem 2. Choose a trangulaton K of P such that ts closed stars form a closed coverng whch refnes V. Snce paracompact perfectly normal spaces are heredtarly paracompact, we can apply Lemma 2 to X = (X, p +1 ), X, p, U and f : U K. We thus obtan open sets U X and mappngs h : U K such that (3.13) (3.16) hold. The frst two of these relatons enable us to apply Lemma 3 and obtan open sets G X such that (3.28), (3.29), and (3.30) are fulflled.

10 348 SIBE MARDEŠIĆ We wll now defne, by nducton on, a sequence of mappngs g : U K. Consder the open sets V 1 =, (3.38) V j+1 = G j+1 jj+1 (U j), j N, and note that (3.39) jj+1 (G j) V j+1 V j+1 U j+1, j N, For = 1 put g 1 = h 1. Assume that we have already defned mappngs g 1,..., g and that they satsfy the followng condtons. (3.40) g j+1 jj+1 (G j) = g j p jj+1 jj+1 (G j), 1 j <, (3.41) g j (U j \V j ) = h j (U j \V j ), 1 j, (3.42) g j+1 (U j+1 \ jj+1 (V j)) h j+1 (U j+1 \ jj+1 (V j)), 1 j <. To defne g +1 apply Lemma 4 to the sets from (3.39) (for j = ) and to the mappngs h +1 : U +1 K and h p +1 V +1. Note that the latter mappng s defned because V (U ). Moreover, by (3.15), h +1 V +1 h p +1 V +1. One obtans a mappng k +1 : U +1 K such that (3.43) k (G ) = h p (G ), (3.44) k +1 (U +1 \V +1 ) = h +1 (U +1 \V +1 ). (3.45) k +1 h +1. Now note that V G U. Furthermore, by (3.41), g (U \V ) = h (U \V ) and therefore, (3.46) g p +1 ( +1 (U )\ +1 (V )) = h p +1 ( +1 (U )\ +1 (V )). We defne g +1 : U +1 K by the formula { g p (3.47) g +1 (x) = +1 (x), x +1 (G ), k +1 (x), x U +1 \ +1 (V ). Note that the sets +1 (G ) and U +1 \ +1 (V ) are closed subsets of U +1 and ther ntersecton S s contaned n the set +1 (U )\ +1 (V ). Therefore, by (3.46), g p +1 S = h p +1 S. On the other hand, S +1 (G ). Therefore, (3.43) shows that also k +1 S = h p +1 S. Consequently, the mappng g +1 s well defned. Clearly, (3.40) holds because of the frst lne n (3.47). In order +1 (G ) V +1 and thus, U +1 \V +1 to verfy (3.41), note that +1 (V ) U +1 \ +1 (V ). Therefore, (3.44) and (3.47) show that (3.41) holds also for j = + 1. Fnally, (3.42) holds for j = because of (3.45) and (3.47).

11 EXTENSION DIMENSION OF INVERSE LIMITS 349 We now defne the mappng g : U K by puttng (3.48) g (G ) = g p (G ), N. Snce p = p +1 p +1, t follows that (3.49) (G ) = +1 (p 1 +1 (G )) +1 (G +1) However, by (3.40), g +1 p +1 (G ) = g p (G ) and thus, (3.50) g +1 p +1 (G ) = g p (G ), whch shows that g s well defned on U = (G ). Contnuty of g s a consequence of the fact that g on (G ) s gven by the contnuous mappng g p and the sets (G ), N, form an open coverng of U. It remans to prove that, for every x U, the ponts f(x) and g(x) belong to a closed star of K. By (1.2) and (3.48), t suffces to prove that, for x (G ), the ponts f(x) and g p (x) belong to a closed star of K. We wll prove ths asserton by nducton on. The asserton s true for = 1, because g 1 (p 1 (x)) = h 1 (p 1 (x)) and, by (3.16), f (U 1 ) h 1 p 1 (U 1 ). Let us now prove the asserton for + 1 assumng that t holds for. By (3.47), g +1 p +1 (x) equals g p (x) or k +1 p +1 (x). In the frst case, the nducton hypothess mples that f(x) and g p (x) belong to a closed star of K. In the second case, (3.45), the ponts k +1 p +1 (x) and h +1 p +1 (x) belong to a closed smplex of K. However, by (3.16), f(x) and h +1 p +1 (x) also belong to a closed smplex of K. Consequently, k +1 p +1 (x) and f(x) belong to a closed star of K. 4. Proof of the man theorem We shall see that Theorem 2 essentally reduces the proof of Theorem 1 to the followng theorem. Theorem 3. Let X = (X, p +1 ) be an nverse sequence of paracompact perfectly normal spaces wth lmt X and surjectve projectons p : X X. Let P be a polyhedron, let A X be a closed set and U X an open set, A U, and let g : U P be a mappng whch admts a fltered factorzaton through X. If dm X P, for every N, then there exsts a mappng h: X P, whch extends g A. Proof of Theorem 3. Let a fltered factorzaton of g : U P be gven by open sets G X and by mappngs g : G P, whch satsfy the analogues of (3.6) (3.8). Consder the open set V = X\A and let V X be the maxmal open set for whch (V ) V. Note that (4.51) +1 (V ) V +1, (4.52) (V ) = V.

12 350 SIBE MARDEŠIĆ An applcaton of Lemma 3 to V and V yelds open sets H X such that (4.53) H V, (4.54) +1 (H ) H +1, (4.55) (H ) = V. Note that, for every, (4.56) p (A) H =. Indeed, snce H s an open set, p (A) H mples that also p (A) H and thus, A (H ) A V, whch s a contradcton. We wll now defne, by nducton on, a sequence of closed sets C X and a sequence of mappngs h : C P, whch have the followng propertes. (4.57) H C G H, (4.58) G p (A) Int C, (4.59) +1 (C ) C +1, (4.60) h (C ) = h p (C ), (4.61) h (C \H ) = g (C \H ). We begn the nducton by puttng C 1 = G 1 H 1. We defne h 1 on G 1 by h 1 G 1 = g 1. We then extend t to C 1 usng the fact that dm X 1 P and thus also dm C 1 P. Now assume that we have already defned the sets C 1,..., C and the mappngs h 1,..., h. In order to defne C +1, note that (4.54) and (4.56) (for + 1) yeld +1 (H ) p +1 (A) =. Therefore, one can fnd an open set W X +1 such that (4.62) p +1 (A) W, (4.63) W X +1 \ +1 (H ). Put (4.64) C +1 = +1 (C ) (G +1 W ) H +1. Cleary, (4.59) s fulflled. (4.57) for + 1 s a consequence of (4.57) for, of (4.54) and of the analogous relaton for G and G +1. Furthermore, (4.58) holds because (4.65) G +1 p +1 (A) G +1 W Int C +1. We defne h +1 on +1 (C ) by (4.60). We also put (4.66) h +1 (G +1 W ) = g +1 (G +1 W ).

13 EXTENSION DIMENSION OF INVERSE LIMITS 351 In order to verfy that (4.66) s compatble wth (4.60), we wll show that both formulas on the ntersecton (4.67) S = +1 (C ) (G +1 W ) yeld the same mappng g p +1 S. Indeed, +1 (H ) (G +1 W ) +1 (H ) W = and thus, S +1 (G ) (G +1 W ) +1 (G ). By Remark 2, g (G ) = g p (G ). Therefore, (4.66) yelds h +1 S = g +1 S = g p +1 S. On the other hand, by (4.63), S +1 (C )\ +1 (H ) = +1 (C \H ) and (4.68) h p (C \H ) = g p (C \H ), because p +1 ( +1 (C \H )) C \H and by (4.61), h (C \H ) = g (C \H ). Hence, defnton (4.60) yelds h +1 S = h p +1 S = g p +1 S. Fnally, dm C +1 P, because dm X +1 P. Therefore, h +1 extends to all of C +1. To verfy (4.61) for + 1, note that (4.69) C +1 \H +1 ( +1 (C )\H +1 ) (G +1 W ). By (4.66), h +1 concdes wth g +1 on the second summand. Now note that, by (4.60), h (C ) = h p (C ). Snce +1 (H ) H +1, we see that (4.70) +1 (C )\H (C \H ) +1 (G ). However, by (4.61) for, we conclude that on the frst summand h p +1 concdes wth g p +1. On the other hand, g (G ) = g p (G ) and thus, (4.70) shows that on the frst summand g +1 also concdes wth g p +1. Now note that (4.57) and (4.58) mply (4.71) (Int C ) = X. Indeed, (4.55) and H Int C mply that the left sde of (4.71) contans V = X\A. Moreover, snce A U, (4.58) and (1.2) mply that the left sde of (4.71) also contans A. We now defne a mappng h: X P by puttng h(x) = h p (x), for x (C ). Notce that x (C ) mples p +1 (x) +1 (C ) and thus, by (4.60), h +1 p +1 (x) = h p +1 p +1 (x) = h p (x), whch shows that h s well defned. It s a contnuous mappng, because t s gven by contnuous mappngs on the open sets (Int C ) whch form a coverng of X. It remans to prove that h A = g A. Frst note that, by (4.53) and (4.52), (H ) (V ) V = X\A and thus, H p (A) =. Moreover, by (4.58), G p (A) C. Therefore, G p (A) C \H. Consequently, by

14 352 SIBE MARDEŠIĆ (4.61), h (G p (A)) = g (G p (A)). Let a A be an arbtrary pont. There s an such that a (G ) and thus, p (a) G p (A). By (4.58), we conclude that p (a) C and thus, h(a) = h p (a). Snce h (G p (A)) = g (G p (A)), we see that h(a) = g p (a). On the other hand, a (G ) mples that also g(a) = g p (a). Consequently, h(a) = g(a). We precede the proof of Theorem 1 by a smple techncal lemma. Lemma 5. Let X = (X, p +1 ) be an nverse sequence of stratfable spaces wth lmt X. Then there exsts a sequence X = (X, p +1 ) wth lmt X and surjectve projectons p : X X such that X s the drect sum of X and of a dscrete space D, the mappng p +1 X +1 = p +1 and the natural ncluson X X embeds X as a closed subset of X. Proof. We construct X and p +1 by nducton on begnnng wth X1 = X 1. By defnton, X+1 = X +1 D +1, where D +1 s a dscrete space whch admts a surjecton D +1 (X \p +1 (X +1 )) D. If a pont x X does not belong to X, then there s an for whch p (x ) D and therefore, t has a neghborhood whch msses X. Hence, X s closed n X. Proof of Theorem 1. We wll frst prove the asserton under the addtonal assumpton that the projectons p : X X are surjectve. Consder a closed set A X and a mappng f : A P. We must show that f extends to all of X. Note that X s stratfable and therefore, the homotopy extenson theorem apples. Consequently, t suffces to produce a mappng g : A P, whch s homotopc to f and extends to all of X. Choose an open coverng V of P such that any two V-near mappngs nto P are homotopc. It suffces to fnd a mappng g : A P, whch s V-near to f and extends to all of X. Snce polyhedra are ANE s for stratfable spaces, the mappng f extends to an open neghborhood U of A, f : U P. Ths enables us to apply Theorem 2 and obtan a mappng g : U P such that g and f are V-near and g admts a fltered factorzaton through X. However, Theorem 3 mples that g A extends to all of X. The case of arbtrary projectons p s reduced to the case of surjectve projectons usng Lemma 5. Indeed, members X of the sequence X are stratfable and dm X P. Snce the projectons p are surjectve, the already establshed case of the theorem yelds the concluson that dm X P. However, X s a closed subset of X, and therefore, the latter relaton shows that also dm X P. Corollary 1. If X = (X, p +1 ) s an nverse sequence of polyhedra X of dmenson dm X n, then the lmt X has dmenson dm X n. Remark 5. In the proof of Theorem 1 for metrzable spaces [18] the frst axom of countablty played an mportant role. In general, stratfable spaces

15 EXTENSION DIMENSION OF INVERSE LIMITS 353 do not satsfy that axom. An easy example s gven by a smplcal complex whch conssts of nfntely many 1-smplexes extng out of one common vertex. Theorem 4. Let P be a polyhedron and let X = (X, p +1 ) be an nverse sequence of stratfable spaces wth lmt X. If dm X P, then also dm X P. The proof of Theorem 4 s a varaton of the proof of Theorem 1. Frst note that dm X P mples dm X P. Therefore, t suffces to consder the case when the projectons p : X X are surjectve. Ths enables us to reduce the problem of extendng f : A P to all of X to the problem of extendng g A to all of X, where g : U P s a mappng whch s defned on an open neghborhood U of A and admts a fltered factorzaton through X. In other words we need the followng varaton of Theorem 3. Theorem 5. Let X = (X, p +1 ) be an nverse sequence of paracompact perfectly normal spaces wth lmt X and surjectve projectons p : X X. Let P be a polyhedron, let A X be a closed set and U X an open set, A U, and let g : U P be a mappng whch admts a fltered factorzaton through X. If dm X P, then there exsts a mappng h: X P, whch extends g A. The proof of Theorem 5 s a varaton of the proof of Theorem 3. In partcular, the sets V, V and H are defned as n the prevous case. One then defnes, by nducton on, an ncreasng sequence of ndces l() N, a sequence of closed sets C l() X l() and a sequence of mappngs h l() : C l() P whch satsfy the analogues of (4.57) (4.61), where has been replaced by l() and + 1 by l( + 1). To begn the nducton we consder the mappng g 1 : G 1 P. Snce dm X P, there exsts an ndex l(1) N such that g 1 p 1l(1) 1l(1) (G 1) extends to C l(1) = 1l(1) (G 1) H l(1). The nducton step s obtaned by a smlar varaton of the nducton step n the proof of Theorem 3. As n (4.71), the unon of the open sets (Int C l() ) equals X and one defnes h by puttng h(x) = h p (x), where x ( C l() ). Fnally, one verfes as before that h A = g A. Remark 6. R.Cauty [2] proved that every CW- complex embeds as a retract n a polyhedron. Therefore, for stratfable spaces X, the homotopy extenson property remans vald f one replaces polyhedra P by CW- complexes. It readly follows that for a polyhedron P and a CW- complex Q of the same homotopy type, the propertes dm X P and dm X Q are equvalent. Snce CW- complexes have the homotopy type of polyhedra, one easly concludes that Theorem 1 remans vald f one assumes that P s a CW- complex. An analogous remark apples to Theorem 5.

16 354 SIBE MARDEŠIĆ Remark 7. In the Fall of 1999 Professor Leonard R. Rubn vsted Zagreb and presented hs work wth Phlp J. Schapro to the Topology semnar of the Mathematcs Department. Ths vst gave the orgnal mpetus for the wrtng of the present paper. In the proof of the man results there s some overlap wth deas encountered n the Rubn Schapro proof. References [1] A.V. Arkhangel skĭ, V.I. Ponomarev: Fundamentals of general topology through problems and exercses (russan), Nauka, Moskva, [2] R. Cauty: Sur les sous-espaces des complexes smplcaux, Bull. Soc. Math. France 100 (1972), [3] R. Cauty: Convexté topologque et prolongement des fonctons contnues, Composto Math. 27 (1973), [4] J. Ceder, Some generalzatons of metrc spaces, Pacfc J. Math. 11 (1961), [5] M.G. Charalambous: An example concernng nverse lmt sequences of normal spaces, Proc. Amer. Math. Soc. 78 (1980), [6] A. Chgogdze: Cohomologcal dmenson of Tychonov spaces, Topology and ts Appl. 79 (1997), [7] H. Cohen: A cohomologcal defnton of dmenson for locally compact Hausdorff spaces, Duke Math. J. 21 (1954), [8] T. Dobrowolsk, L.R. Rubn: The hyperspaces of nfnte -dmensonal compacta for coverng and cohomologcal dmenson are homeomorphc, Pacfc J. Math. 164 (1994), [9] A. Dranshnkov: The Elenberg-Borsuk theorem for mappngs n an arbtrary complex (russan), Mat. Sbornk 185 (1994), no. 4, (Transl. Sbornk Mat. 81 (1995), No. 2, ). [10] A. Dranshnkov, J. Dydak: Extenson dmenson and extenson types, Proc. Steklov Inst. Math. 212 (1996), [11] R. Engelkng: General topology, Polsh Scentfc Publshers, Warszawa, [12] R. Engelkng: Dmenson theory, North -Holland, Amsterdam, [13] P.J. Huber: Homotopcal cohomology and Čech cohomology, Math. Ann. 144 (1961), [14] Y. Kodama: Note on cohomologcal dmenson for non-compact spaces, J. Math. Soc. Japan 18 (1966), [15] S. Mardešć, J. Segal, Shape theory, North -Holland, Amsterdam, [16] K. Nagam: Fnte-to-one closed mappngs and dmenson. II, Proc. Japan Acad. 35 (1959), [17] L.R. Rubn: A stronger lmt theorem n extenson theory, Glasnk Mat. (to appear). [18] L.R. Rubn, P.J. Schapro: Lmt theorem for nverse sequences of metrc spaces n extenson theory, Pacfc J. Math. 187 (1999), [19] E.G. Sklyarenko: On the defnton of cohomologcal dmenson (russan), Dokl. Akad. Nauk SSSR 161 (1965), No. 3, (Transl. Sovet Math. Dokl. 6 (1965), ). Department of Mathematcs Unversty of Zagreb P.O.Box Zagreb Croata E-mal: smardes@math.hr