PRODUCTS OF SPACES OF COUNTABLE TIGHTNESS

Transcription

1 Vlume 6, 1981 Pges PRODUCTS OF SPACES OF COUNTABLE TIGHTNESS by Yshi Tk Tplgy Prceedigs Web: Mil: Tplgy Prceedigs Deprtmet f Mthemtics & Sttistics Aubur Uiversity, Albm 36849, USA E-mil: tplg@ubur.edu ISSN: COPYRIGHT c by Tplgy Prceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Vlume PRODUCTS OF SPACES OF COUNTABLE TIGHTNESS Yshi Tk Itrducti As is well kw, the prduct X f spce X f cutble tightess eed t hve cutble tightess. Als if X is CW-cmplex, x is t lwys CW-cmplex. I this pper, i the first secti, we csider the prducts f spces f cutble tightess d k-spces. I the secd secti, we csider the prducts d the metrizbility f CW-cmplexes. 1. Prducts f k-spces d Spces fcutble Tightess All spces re ssumed t be regulr' d T. We cl sider crdils t be iitil rdils, d let c dete the crdility f the ctiuum. Let N be the set f turl umbers. We eed the fllwig well kw exmple. This exmple will ply imprtt rle i the prducts. Let be ifiite crdil umber. Let S be the spce btied frm the disjit ui f cverget sequeces by idetifyig ll the limit pits. S is w especilly clled the sequetil f. We w recll sme bsic defiitis. Let X be spce, d let J = {F y : y E r} be clsed cverig f X. The X hs the wek tplgy with respect t J, if F c X is clsed wheever F F y is clsed i X fr ech y E r.

3 116 Tk A spce X is k-spce (resp. sequetil spce), if X hs the wek tplgy with respect t the cllecti f ll cmpct subsets (resp. cmpct metric subsets) f X. A spce X is kw-spce [11], if it hs the wek tplgy with respect t cutble cverig f cmpct subsets f X. A spce X hs cutble tightess, t(x) ; W, if x E A i X, the x E C fr sme cutble C c A. It is kw tht every sequetil spce hs cutble tightess. Prpsiti 1.1. (1) If X x Sc is k-spce, the ech clsed, seprble subset f X is lclly cutbly cmpct. () If X x Sc hs cutble tightess, the ech k -subspce f X is lclly cmpct. w Prf. (1) Suppse tht there exists clsed, seprble subset S f X which is t lclly cutbly cmpct. Sice S is regulr d T l, s is well kw, the weight f S is equl r less th c. Hece sme X E S hs lcl bse {U: < m} i S, w < m < c, such tht ech U is t cutbly cmpct. We w use the ide f E. Michel [10; Therem.1]. Fr < m, sice U is t cutbly cmpct, there is decresig sequece {F ; E N} f -empty clsed subsets f U with F =~. Let T = U{F x ; EN}, where detes the -th term f the -th sequece i 8, d m let T UT. The fr ech cmpct subset K f S x Sm' <m T K is clsed i S x Sm' becuse K meets ly fiitely is fiite ui f clsed sub my Tts d ech K T sets f S x S. But T is t clsed i S x S. This m m

4 TOPOLOGY PROCEEDINGS Vlume implies tht S x Sm is t k-spce. Sice S x S is m clsed subset f X x Sc' X x Sc is t k-spce. ctrdicti. This is () If spce hs cutble tightess, s des every subspce. Thus we my ssume tht X is kw-spce. Sice t(x x Sc) ~ w, X x Sc hs the wek tplgy with respect t the cverig f ll clsed seprble subsets f X x Sc. Sice ech subset S f X x S is ctied i X x TI(S), c where TI: X x Sc ~ Sc is the prjecti, X x Sc hs the wek tplgy with respect t clsed cverig {X x F; F is clsed seprble subset f Sc}. Sice we c ssume tht ech F is ctied i sme S' < w l ' F is kw-spce. By [11; (7.5)], ech X x F is k-spce. Thus X x S is c k-spce. Hece, by (1) ech clsed, seprble subset f X is lclly cutbly cmpct. We w shw tht X is lclly cmpct. Let X hve the wek tplgy with respect t cutble cverig f cmpct subsets Xi with Xi c X i + l. Fr sme X E X, suppse X E X - Xi fr ech i. Sice t(x) ~ w, there re cutble Let C = U~ lc,. The 1= 1 X E C (X - Xi) fr ech i. Sice the clsed seprble subset C f X is lclly cutbly cmpct, there exists cutbly cmpct subset K f C such tht X E K (X - Xi) fr ech i. Sice K is cutbly cmpct i X, it is esy t see tht K is ctied i sme X.. But 1 x E K (X - X. ) = ~. This is ctrdicti. Thus 1 0 ech pit f X is ctied i sme it Xi. Hece X is lclly cmpct.

5 118 Tk A spce X is strgly Frechet [14], i.e. cutbly bi-sequetil due t E. Michel [1], if x E X- with A + l ~ A the there exist x E A such tht x ~ x. If the A re ll the sme set, the such spce X is Frechet. Lemm 1.. (cf. [15; 16 (b) d p. 35]). Every Frechet spce which is t strgly Frechet ctis cpy Recll tht spce X is symmetric if there is rel vlued, -egtive fucti d defied X x X stisfyig the cditis: (1) d(x,y) = wheever x = y; () d(x,y) d (y, x) ; d (3) A c X is clsed i X wheever d(x,a) > 0 fr y x E X-A. If we replce the cditi (3) by the fllwig: Fr A ~ X, x E X if d ly if d(x,a) = 0, the such spce is clled semi-metric. Crllry 1.3. Suppse X x Sc hs cutble tightess. (I) If X is Frechet, the X is strgly Frechet. () [CH]. If X is symmetric, the X is semi-metric. Whe X is prcmpct, [CH] c be mitted. Prf. (1) This fllws frm Prpsiti 1.1() d Lemm 1.. () Let X be symmetric spce. Every Frechet d symmetric spce is first-cutble ([1; p. 19]), hece is semi-metric. S, we prve tht X is Frechet. T prve this, sice t(x) ~ w, it is sufficiet t shw tht every clsed, seprble subset S f X is first cutble. Sice

6 TOPOLOGY PROCEEDINGS Vlume S is regulr d T ech pit f S hs lcl bse f l, crdility c i S. The, uder CH ech pit f S is G 8 -set i S by [16; Therem 10]. Whe X is prcmpct spce, withut [CH], the seprble spce S is Lidelf. Thus, by [13; Therem ] S is hereditrily Lidelf. The ech pit f S is G 8 -set i S. ech pit f S is G 8 -set i S. Hece, the i y cse Thus, by Prpsiti 1.1() d [8; Lemm 6.11], S is first cutble. A bi-k-spce, ccrdig t E. Michel [1], is chrcterized s bi-qutiet imge f prcmpct M-spce. Fr the itrisic defiiti f bi-k-spce, see [1; Defiiti 3.E.l]. Crllry 1.4. Suppse f: X ~ y is clsed mp with t(y) ~ w. Let X be prcmpct bi-k-spce (resp. prcmpct lclly cmpct spce). The Y x Sc is k-spce (resp. t(y x Sc) < w) if d ly if Y is lclly cmpct. Prf. Let Y be lclly cmpct. The.Y x Sc is k-spce (resp. t(y x Sc) ~ w) by [3; 3.] (resp. [9; Therem 4]. S we prve the "ly if" prt. Suppse Y x Sc is k-spce. The, by Prpsiti 1.1(1), Y hs prperty (P): Every clsed seprble subset is lclly cutbly cmpct. The, sice t(y) ~ w, it is esy t see tht Y stisfies Lemm 9.l(b) i [1]. Ideed, if {F : E N} is decres ig sequece with y E (F - {y}), the there exist Y E F such tht {Y: E N} is t clsed i Y. The, by [1; -1 Therem 9.9], ech f (y) is cmpct. Thus, by [1; Prpsiti 3.E.4], Y is bi-k-spce.

7 10 Tk Next, we prve tht Y is lclly cmpct. Suppse t. The there is pit Y E Y such tht Y E ~ fr every cmpct subset K f Y. Let J = {X - K; K is cmpct i y}. The J is filter bse ccumultig t the pit Y. Sice Y is bi-k, by [1; Lemm 3.E.] there is decresig clsed sequece {A: E N} stisfyig the fllwig: () C = A is cmpct; (b) If V is pe subset f Y with C c V, the C c A ~ V fr sme ; d (c) Y E F A fr ll E N d ll F E J. T prve sme A is cmpct, suppse t. prcmpct, ech A is t c6utbly cmpct. Sice Y is The there re clsed discrete subsets D f A with ID I = w. Let Y c U U D be subspce f Y. The Y is =l clsed i Y. Let Z be qutiet spce btied frm Y by idetifyig the cmpct subset C. is t lclly cutbly cmpct. The, by () d (b), Z Sice Y stisfies (P) d the cutble spce Z is the perfect imge f clsed seprble subset f Y' s the Z is lclly cutbly cmpct. This is ctrdicti. Hece sme A is cmpct. But, by (c~ Y E F N =~. This is ctrdicti. Hece Y is lclly cmpct. Filly we prve the pretheticl prt. Let t(y x Sc) ~ wd let T be y clsed seprble subset f Y. The T is clsed imge f clsed seprble subset S f X. Sice X is prcmpct, S is Lidelf. Sice X is lclly cmpct, it is esy t see tht S is kw-spce.

8 TOPOLOGY PROCEEDINGS Vlume Thus, sice T is qutiet imge f S, T is ls kw-spce. The, by Prpsiti 1.1(), T is lclly cmpct. Hece Y hs Prperty (P). Thus, sice t(y) ~ w, Y stisfies Lemm 9.l(b) i [1]. df-l(y) is cmpct. S, by [1; Therem 9.9] ech Thus Y is lclly cmpct. Let be ifiite crdil. Recll tht spce X is -cmpct if every subset f X f crdility hs ccumulti pit i X. Lemm 1.5. Let f: X ~ Y be clsed mp with X cllectiwise rml d Y sequetil. If Y ctis clsed cpy f S"' the ech df-l(y) is -cmpct. Prf Suppse sme df-l(y ) is t -cmpct. The 0 there exists clsed discrete subset D f df-l(y ) with 0 IDI =. Hece there is discrete pe cllecti {V d ; d E D} f X with V 3 d. Fr ech d E D, sice d Y E f(v ) - f(v d ). {Y}' Y is t islted i sequetil spce S the there is sequece Cd = {Yd; E N} such tht Y ~ Y d Cd ~ f(v d ) - {Y} Sice {f(v d ); d E D} is hereditrily clsure preservig, s is the cllecti ( = {Cd U {Y}; d ED}. Let Y be the ui f (. The Y is clsed i Y. Let Z be the disjit ui f [, d let g: Z ~ Y be the bvius mp. The Z is metric d g -1 is clsed with dg (Y) t -cmpct. Hece, by [7; Lemm ], Y ctis clsed cpy f S. Thus Y ctis clsed cpy f S. This is ctrdicti. Frm Prpsiti 1.1() d Lemm 1.5, we hve

9 1 Tk Crllry 1.6. Let f: X ~ Y be clsed mp with X prcmpct sequetil. If t(y x Sc) ~ w~ the ech -1 f (y) is cmpct. By Lemm 1.5, we c geerlize ll results i this secti s fllws. Geerlizti. Let S be sequetil spce which is clsed imge f cllectiwise rml spce uder f -1 such tht sme f (s) is t c-cmpct. The, fr ll results i this secti we c replce "Sc" by "S." By this geerlizti, fr exmple we hve the fllwig: Let Y be Frechet spce. Let X be cllectiwise rml sequetil spce, d let F be clsed subset f X. Suppse Z is qutiet spce btied frm X idetifyig F. The Y is strgly Frechet r F is c-cmpct, if t (Y x Z) : w.. CW-Cmplexes The ccept f CW-cmplexes due t J. H. C. Whitehed [17] is well-kw. We recll sme bsic prperties f CW-cmplexes. Let X be CW-cmplex; tht is, X is cmplex which is clsure fiite (i.e. ech cell f X is ctied i fiite subcmplex), d which hs the wek tplgy with respect t the clsed cverig {Lyi Y E f} f ll fiite subcmplexes L f X. Y The fr y subset f' f f, L' = U L is clsed i X d L' hs the wek tplgy YEf' y with respect t clsed cverig {Lyi y E f' }.

10 TOPOLOGY PROCEEDINGS Vlume As tplgicl cmplex, C. H. Dwker [4] itrduced the ccept f the Whitehed cmplex. A spce X is Whitehed cmplex, if it is ffie cmplex (see [4~ l]) hvig the wek tplgy with respect t {ea~ A}. Here {ea~ A} is the cells f X. Recll tht the clsure e A f e cicides with the tplgicl clsure i X f e A A [4~ p. 560], d this ls hlds i CW-cmplexes. Every Whitehed cmplex with the cells {ea~ A} is CW-cmplex with ech e A subcmplex [4~ p. 558]. We eed the cicl exmple S due t S. P. Frkli [5~ Exmple 5.1]. Tht is, S = (N x N) U N U {J with ech pit f N x N is islted pit. A bsis f eighbrhds f E N csists f ll sets f the frm {} U {(m,)~ m ~ mol. Ad U is eighbrhd f 0 if d ly if 0 E U d U is eighbrhd f ll but fiitely my E N. Lemm.1. Suppse tht X hs the wek tplgy with respect t pit-cutble clsed cverig {C ~ } f x. (1) Let ech C be Frechet. The X is Frechet if d ly if X ctis cpy f 5. () Let ech C be metric. The X is metric if d ly if X is prcmpct~ strgly Frechet spce. Prf. (1) Sice S is t Frechet, the "ly if" prt fllws frm tht every subset f Frechet spce is Frechet. We prve the "if" prt. Sice X is sequetil, by [5~ Suppse X is t Frechet. Prpsiti 7.3] X ctis

11 14 Tk subspce M = (N x N) U N U {J which, with the sequetil clsure tplgy, is cpy f 5' The cutble spce M itersects t mst cutbly my C's, sy C,... l Let X U C d let C be cmpct subset f M. The i=l i C hs the wek tplgy with respect t cutble clsed cverig {X C; E N} f C. Hece C is ctied i sme X c. Thus ech cverget sequece i M is c tied i sme X. We ls remrk tht ech X hece ctis cpy f M. is Frechet, We w use the methd f prf f S. P. Frkli d B. V. Smith Thms [6; Prpsiti 1). Sice N U {OJ is cverget sequece i M, there is X with N U {OJ ~ X Let C = {} x N U {} fr ech. Sice C is cver l get sequece, there is X ( > ) with C ~ X l l l l Sice X is clsed d Frechet, we c chse C ( > 1) l d X ( > ) such tht C X is t mst fiite d 3 3 l C c X S, we c ssume tht C c X - X I 3 3 l this wy, we c chse C d X ( + > > _ ) k l k k l k k + l 00 Let M' = U C U { ; ken} U{O}. k=l k k The, fr ech E A, M' C is clsed i X. Thus M' is clsed subset f X. The M' is sequetil, hece M' hs the sequetil clsure tplgy. Thus M' is cpy f 5' Hece X ctis cpy f 5' This is ctrdicti. () We prve ly the "if" prt. Fr x E X let { C ; C 3 x} be {C C } Put X U C Suppse Ci l i=l i

12 TOPOLOGY PROCEEDINGS Vlume x E X - X fr ech. Sice X is strgly Frechet, there exist x X such tht x ~ x. Let C {x i E N} U {x}. The the cmpct subset C hs the wek tplgy with respect t cutble cverig {C C i C C 1.0} f C. The C is ctied i sme ex. ex. fiite ui f C Thus sme C must cti ifiitely 0 my x's, hece C :3 x. The C is ctied i sme 0 0 X But this is ctrdicti, fr X -::h X fr ~ -? - 0 Thus x X - X fr sme, hece x E it X. This implies tht X is lclly metrizble. X is prcmpct. Hece X is metrizble, fr Lemm.. Let X be CW-cmplex with the cells {e y }. If X ctis clsed cpy f S~ the fr ech x E X the crdility f r = {Yi e y x 3 x} is less th. Prf. Fr sme X E X, suppse Ifx I ~. Sice e E x fr Y E r, there exist x such tht x ~ x d Y 0 x y y 0 0 x E e Let C = {x E N} U {x } d let S = U{C yi y y y y' 0 y E r }. Suppse L is y fiite subcmplex f X. The x 0 S L is clsed i x. Thus S is clsed i x. Mrever S hs the wek tplgy with respect t {C i Y E r }. Ideed, y X fr F c S, let F c be clsed i S fr ech y E f The y X F L {F c i eel d y E r }. Thus F L is clsed y y X i S. Hece, F is clsed i S. This implies tht X ctis clsed cpy f S. This is ctrdicti. I [6], S. P. Frkli d B. V. Smith Thms prved tht kw-spce with metrizble "pieces" is metrizble if

13 16 Tk d ly if it ctis cpy f 8 d sequetil f 8 w Algusly t this result, we hve Prpsiti.3. Let X be () CW-cmplex (resp. Whitehed cmplex)~ r (b) prcmpct spce hvig the wek tplgy with respect t pit-cutble clsed cverig f metric spces. The the fllwig re equivlet. f 8 ). (1) X is metrizble. () X ctis cpy f S d Sw (resp. cpy (3) t(x x 8 ) ~ w. c Prf (1) ~ () is esy. We hve (3) ~ () frm Prpsiti 1.1. (). (1) ~ (3) fllws frm [; Crllry 4] () ~ (1). I cse f (b), we hve this implicti frm Lemms 1. d.1. S, we suppse X is CW-cmplex. First we prve tht X is Frechet. T see this, sice t(x) ~ w, it is sufficiet t shw tht every clsed seprble subset S = IT with D cutble, is Frechet. Clerly, D is ctied i sme cutble ui L f fiite subcmplexes L. Sice L is clsed i X, S is clsed subset f L. Thus S hs the wek tplgy with respect t cutble cverig f cmpct metric subsets L S f S. Sice S ctis cpy f S' by Lemm.1(1), S is Frechet. The X is Frechet. Secd we prve tht X is metrizble. Sice X ctis cpy f Sw' by Lemm., X hs the cells {e.:\} such tht

14 TOPOLOGY PROCEEDINGS Vlume {e }, e = cl e, is pit fiite. Fr x E X, let A A A Put 'E = U e i=l A i Suppse x E x-=-e. Sice X is Frechet, there is cverget sequece {x ; E N} such tht x + x d x ~ E. Sice the cverget sequece is ctied i fiite ui f cells e, sme ea. must cti ifiitely my x's. A 1 0 Hece x E ea. Thus e = ea. fr sme i O ~. But this is A 1 ctrdicti, becuse x ~ e A fr ll. The x ~ x-=-e, which implies x E it E. Sice E is cmpct metric, X is lclly metrizble. The X is metrizble, fr X is prcmpct. Sice pit-fiite Whitehed cmplex is lclly fiite, the prethetic prt is prved similrly. Let I be the spce btied frm disjit ui f clsed uit itervls [0,1] by idetifyig ll zer pits. The ech I is Whitehed cmplex. C. H. Dwker [4] shwed tht I x I is t Whitehed cmplex. w c Frm Prpsiti.3 d Lemm., we hve the fllwig geerlizti ~ the Dwker's exmple. Crllry.4. Let X x Y be CW-cmplex d {e ; A} A be the cells f Y. The X is metrizble~ r ech crdility f {A; e 3 y} is less th c. A Prpsiti.5. Suppse tht Xl d X re CW-cmplexes (resp. Whitehed cmplexes). The the fllwig re equivlet. (1) t(x x X ) < w. l () Xl x X is k-spe.

15 18 Tk (3) Xl x X is CW-cmplex (resp. Whitehed cmplex). Prf (1) ~ (). Sice t(x x X ) ~ w, Xl x X hs l the wek tplgy with respect t the clsed cverig f ll clsed, seprble subsets f Xl x X. Ech subset S f Xl x X is clerly ctied i ITl(S) x IT (S), where IT i : Xl x X ~ Xi (i = 1,) re prjectis. Thus Xl x X hs the wek tplgy with respect t cverig {F x F ; l F i is clsed seprble subset f Xi}. As is see i the prf f Prpsiti.3, () ~ (1), ech F is kw-spce. i Hece, by [11; (7.5)] ech F l x F is k-spce. This implies Xl x X is k-spce. () ~ (3). Let {e }; {e } be the cells f Xl; X y 8 respectively. Sice Xl d X re cmplexes; ffie cm plexes, Xl x X is cmplex; ffie cmplex with cells {e x e } respectively. Mrever, if Xl d X re y 8 CW-cmplexes, the Xl x X is clsure fiite. Thus, t prve tht Xl x X plex), we ly shw tht Xl x X respect t cllecti {e y x e}. rs CW-cmplex (ls, Whitehed cm hs the wek tplgy with Ech cmpct subset f Xl x X is ctied i cmpct subset f Xl x X with type A x B. The, ech cmpct subset f Xl x X is c tied i fiite ui f e x e 8 Sice Xl x X is y k-spce, this implies tht Xl x X with respect t the cllecti {e x e}. y We hve (3) ~ sequetil, hece t(x x X ) ~ w. l hs the wek tplgy (1) frm tht every CW-cmplex is Let X be CW-cmplex with the cells {e}. The we y shll cll X pit-fiite; pit-cutble; lclly

16 TOPOLOGY PROCEEDINGS Vlume cutble, if the cverig{e } f X is s respectively. Y Lemm.6. Let X be Frechet CW-cmplex r Whitehed cmplex. If X is pit-cutble~ the it is lclly cutble. Prf. Sice every pit-cutble Whitehed cmplex is lclly cutble, the we suppse tht X is Frechet CW-cmplex. Let {e } be the cells f X such tht {e } is Y y pit-cutble. Fr x E X, let {e e 3 x} be {~, y' Y Yl - e,... }. Put E U e Sice X is Frechet, by the Y y. i=l 1 prf f Prpsiti.3, () ~ (1), we hve x x-=-e. This implies x E it E. Sice ech e is cmpct, by the Yi prf f [17; (D)], ech e Y i meets t mst fiitely my ey's, s tht it E meets t mst cutbly my e IS.. Y This implies tht X is lclly cutble. prt is prved similrly. The prethetic Prpsiti.7. Whitehed cmplex). Let X be Frechet CW-cmplex (resp. The the fllwig re equivlet. (1) X is pit-cutble. () X is lclly cutble. (3) x is CW-cmplex (resp. Whitehed cmplex). Prf (1) ~ () fllws frm Lemm.6. () ~ (3). Every lclly cutble CW-cmplex is kw-spce, d every prduct f tw lclly kw-spces is k-spce. Thus () ~ (3) fllws frm Prpsiti.5. (3) ~ (1). Suppse tht X is t pit-cutble. The, by Lemm., X ctis clsed cpy f S wl

17 130 Tk Thus X is k-spce which ctis clsed cpy f s. wi Hece S is k-spce. Hwever, by [7; Lemm 5], 8 is wi WI t k-spce. This is ctrdicti. I terms f set-theretic xim BF(w ) belw, we shll csider the prduct X x Y f CW-cmplexes X d Y. BF(w ): If F ={f; f: N + N is fucti} hs crdility less th w ' the there is fucti g: N + such tht { E N; f() > g()} is fiite fr ll f E F. Hece CH implies BF(W ) is flse. I [7], Gry Gruehge prved the fllwig result (*): (*) S x 8 is k-spce if d ly if BF(W ) hlds. W WI Frm this result (*), if the sserti f Prpsiti 1.1 by replcig "S " by "s "hlds, the BF(W ) is flse. c WI Lemm.8. I x I is Whitehed cmplex if d ly W WI if BF(W ) hlds. Prf. "If." Sice BF(W ) hlds, by the prf f [7; Lemm 1] it turs ut tht I x I is sequetil. W WI Hece I x I is Whitehed cmplex by Prpsiti.5. W WI "Oly if." I W x I is k-spce d it ctis clsed WI cpy f Sw x 8, s tht 8 x 8 is k-spce. Thus by WI W WI the result (*), BF(W ) hlds. N Prpsiti.9. If X d Yre Frechet CW-cmplexes (resp. Whitehed cmplexes)~ the the fllwig re equivlet. (1) X x Y is CW-cmplex (resp. Whitehed cmplex)

18 TOPOLOGY PROCEEDINGS Vlume if d ly if X r Y is lclly fiite, therwise X d Y re lclly cutble. () BF(W ) is flse. Prf (1) ~ () fllws frm Lemm.8. () ~ (1). The "if" prt f (1) des t use (). Suppse tht X r Y is lclly fiite CW-cmplex. The X r Y is lclly cmpct. Thus X x Y is k-spce. Suppse tht X d Yre lclly cutble. The they re lclly kw-srces,hece X x Y is k-spce. I y cse, X x Y is k-spce. Hece X x Y is CW-cmplex by Prpsiti.5. The prethetic prt is prved similrly. Next we prve the "ly if" prt. Suppse tht Y is t lclly cutble. The by Lemm.6, Y is t pitcutble CW-cmplex. The by Lemm., Y ctis clsed cpy f S. T shw X is pit-fiite, suppse t. w l The X ctis clsed cpy f Sw by Lemm.. Thus X x Y ctis clsed cpy f S x S. Sice BF(W ) w w l is flse, Sw x S is t k-spce by the result (*). w l But, sice X x Y is CW-cmplex, Sw x S w l is k-spce. This is ctrdicti. Thus X is pit-fiite, hece is lclly fiite by Lemm.6. if X is t lclly cutble. Similrly, Y is lclly fiite This fiishes the prf. The fllwig questis () d (b) remi, the ltter is relted t Prpsiti.7. Questis. () Fr every CW-cmplexes X d Y, des (1) # () f the previus prpsiti hld? (b) Is X lclly cutble if x is CW-cmplex?

19 13 Tk Supplemet Quite recetly, thrugh Zhu H-xu, the uthr lered f the fllwig result due t Liu Yig-mig era ecessry d sufficiet cditi fr the prduct f CW-cmp1exes," Act Mthemtic Siic, 1 (1978), (Chiese). [CH] Let X d Y be CW-cmplexes. The X x Y is CW-cmplex if d ly if either X r Y is lclly fiite, r X d Yre lclly cutble. Referrig t the bve pper d G. Gruehge [7], we c prve tht the swers t the questis () d (b) re ffirmtive. The uthr wishes t thk Zhu H-xu fr his trs 1ti f Liuls pper. Refereces [1] A. V. Arhgellskii, Mppigs d spces, Russi Mth. Surveys 1 (1966), [], The frequecy spectrum f tplgicl spce d the clssificti f spces, Sviet Mth. Dkl. 13 (197), [3] D. E. Che, Spces with wek tplgy, Qurt. J. Mth., Oxfrd Sere 5 (1954), [4] C. H. Dwker, Tplgy f metric cmplexes, Amer. J. Mth. 74 (195), [5] S. P. Frkli, Spces i which sequeces suffice II, Fud. Mth. LXI (1967), [6] d B. V. Smith Thms, O the metrizbility f kw-spces, Pcific J. f Mth. 7 (1977), [7] G. Gruehge, k-spces d prducts f clsed imges f metric spces, Prc. Amer. Mth. Sc. 80 (1980),

20 TOPOLOGY PROCEEDINGS Vlume [8] P. W. Hrley III d R. M. Stephes, Jr., Symmetrizble d relted spces, Trs. f Amer. Mth. Sc. 19 (1976), [9] V. I. M1yhi, O tightess d susli umber i exp X d i prducts f spces, Sviet Mth. Dk1. 13 (197), [10] E. Michel, Lcl cmpctess d crtesi prducts f qutiet mps d k-spces, A. Ist. Furier, Greble 18, (1968), [11], Bi-qutiet mps d crtesi prducts f qutiet mps, A. Ist. Furier, Greble 18, (1968), [1], A quituple qutiet quest, Geerl Tplgy d App1. (197), [13] S. Nedev, Symmetrizble spces d fil cmpctess, Sviet Mth. Dk1. 8 (1967), [14] F. Siwiec, Sequece-cverig d cutbly bi-qutiet mppigs, Geerl Tplgy d App1. 1 (1971), [15], Geerliztis f the first xim f cutbility, Rcky Muti J. f Mth. 5 (1975), [16] R. M. Stephes, Jr., Symmetrizble~ J-~ d wekly first cutble spces, C. J. Mth.. XXIX (1977), [17] J. H. C. Whitehed, Cmbitril hmtpy. I, Bull. f Amer. Mth. Sc. 55 (19A9), Tky Gkugei Uiversity Kgei, Tky, Jp