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(2) . 1 VO ,16 , NQ I. 工A ion . ) Sec i ive t i Un t . l d ido,Gakuge r s l 。f Ho y( { Journa a. Aug .1965. ing Dimens .on 。f the ル=Chae1 Space The Cover. Nobuo KIMURA f MathematicS, Hakodate Branch, The Study 。 ty l Hokkaido Gakugei Univers. l 空間の次元 chae 木 村 信 夫 : Mi. 1ntroduction. f for each 行nite open covering G of x there exists a cal space Let x be a topologi . l l say thatthecovering dimens・on l f G wi th order not greater than n十1 , wesha re負nement o <: f dim X <:n is true but dim X <:n- l is false we of × is not greater than n: dim X n. l ing dimension of x equals to n: dim x =n. l say thatthe cover shal l l be ing” wi ive “cover s possible the adject Throughout thi s paper when no confusion i i t ted. ・ on. cal product X×Y of a space X andaseparable Now, as to the dimension of the topologi ion ic space Y, the relat metr. im Y d im (X×Y)≦ dim X 十d f X×Y is not normal l( see [1]) tandnorma , .l scountablyparacompac holds whenever X×Yi l i h h t f t n h i t t o t r e a h e a l 頓 ′ n a c e s own i a v e , , ) does not hold n genera. on the other hand ,(A im Y dim (X×Y)>dim X 十d (B) f a hereditarily paracompact space X l to olo ical roduct X × Y o h (A). holds for t e non‐norma. p g p lm[ l 2 t d by E コ c ・ae h i Y 心l i . 帆/e have l b c e . Mi , w ch was construc e and a separa e metr 疑 e spa ion of l as the dimens h dime h l lsion of X as We1 ー fated, on that oCcasioー , t at t e also demonst , Y are zero: dim X = dim Y =0. ’ i ioned 1 chaels space l show that the dimension of the above‐ment ln thi s paper we shal. im (X> くY)=1 i , s actually one: d SI .. ] by inedf rom the closed unit interval [0 s obta ,1 Let X be a topologi cal space whにhi * ) ( F 4 G n d th = 十 f n q i wi s expresse as 1is open if and only i l ・at a set N tso t zing i retopologi f l l i ional i s rrat st ー 1g of i th F consi ・ Then the o ow ng }d wi lse al l open set G in the usual sel al lows([1]): ion f t ol proposi tion l Proposi ジ” ×=0, . メメ. li ionals ing of al rrat losed interval [0, 1] consist . Then the fc Let Y be a subspace o. l known ( l e g lowi fol . . [3]) . ng proposition is we. ” ” ” ‘ ’ ins ’ i on” , o denote union and intersect 葵 lemploy ”+“ and‘ l nd (,t ) ln th ead of ) a ( i r wesha , t spape ly i t ve spec re .. (8).
(3) . Nobuo Kimura Proposi tion 2 ・ α多粥 γ= α .he space XXY sha l lbe referred to,from now on, as ハ江icha s space has el space. Thi ‐ been proved to be noルno l([2]) tis not zero‐dimensional, rma , Hence i Proposision 3 Z Z o7 プ ク郷〃s eZ s の Z鋼s zα o” o/ 肌沈みα郷 sPαc , r彰 メメ. α勿2(ズ× Y)≧ヱ, Now the fol lowi ng proposition plays an important role in this paper, r卿 鰯“解郡Z β dogs “oZ B%ceed o解 o“ ひず みのcた解Z 物αc. Proposi tion 4 ,. d Z粥(xx y )≦ 乙 ・ l l rational mlmbers in x and L the set of all irrat ional P箆of . Let R be the set of a 1 lumbers i l l xう then We can express: × コ R + L, Let us denote by( )the ordinary- -in thesense of Euclidean metric---openinterval a ,b. th end points a and b. contained i n [0 ,1] wi. b d k=0 1 ..,,.. 2n f Set , , , .-1, , or any natural num er n an. since. 1 ) (. k. 駆 and. k好 2. ・‐ 儀, 誉 ) ,Y , l numbers low 強 om the de6nmon of Y 由a iona t arerat ,iH ol. 1。 i t s an open and closed subsetin、Y for each n and k, From the de云ni 。n of l。k we ,i. have , for any n =1 ,2 ,.,,..,,. 1 1-1 2. Y= 1 1血 k=O. bt ly given 6 Let {U1 ing o fx×Y. Theni te open cover rari ni t r , U2 , = ...., U.} be an ar i ‐ 伍 h h 五 f ruct an open re nement o {Ui ll≦i≦r} wi su ces to s 。w t at we can const th order not exceeding 2 . Fi ional numbers wh rst We select in x a countable set {の i ch ili=1 ,2 ,,,,,,,} ofirrat. i th respectto the ordinary Euclidean metric, Strictly saying, {のiーiニー s dense Wi ,2 ,.,,..}. i th U( )ヨ のn for any x E R and any neighborhood x s such a set that there exists an n wi. f x. U( )o x l ini ion of the top。logy imposed upon × owing to t t ・e def. 2 ) (の, の ={X1の <×<のt ( , X E X} turns outt。 be open and closed in x,. l Ev ident )1の <のも } Consists ofaCountable numberofintervals,hence wecan arrang仁 y {(の彫 のt them in order: 1 ) J( 2 ) J( , ,.....,,. ) (3. i ) bei ( l lg some (の, の ) each J む ,. ー. Now the f lowi ol ng properties of spaces are quite elementary.. f 。pen and closed subsets i t 4 ni e union o ) Any 6 ( )( a s also 。pen and closed, ‐ i b ) any nite intersection of 。pen and closed subsetsノis also open and closed, (. ) ( c. f two open and c losed subsets i any di畳erenCe o s also open and closed,. d fA is an )i (. losed subset of × and B an open and closed subset o fY then open and c. A×B is open and closed in X×Y. Let us put. 5 ) (. ( )×ふk1J ( )×Lk〔 u 1≦s< 閃 1≦ n< 閃 0≦ k ≦2n一1} s s U′ iニ ヱ{J , , , ,. (9).
(4) . ′ 1Space i i i rhe C0ver Chae S (m d the ハ4 l l g Dime l () C l d X Y t d 4 2 on grounds of( 1 ) )and ( )of( ) ,js ×Lk are open and c ose in X , onsequen y ,(. ions and thei h r 負ni te intersect r di托erences are also oPen and closed, thei te unions r 6ni , t ei. ) and lnk it follows that ( s From the de負nitions of J. 6 ) (. f X×Y covering R×Y. 刃 U′ iis an open subset o h l 5 工 owi ng form: ) n the fol ti s possible to express ( ,i ,for eac i:1≦i≦r. U′ i= 又 V▽ i t , も=1 () k each 帆γ i も being Js ×lnk forsome s , n, .. (5′ ). ion < in t i he set {( ÷ くm} ≦ ≦i≦r l l introduce an ordering relat )11… Now we sha ,1≦≦t ,t lows: as fol. ft<s or t=s andi<i f and only i i e 7 )<( i )i ) ( ( . . ,2) ,1)<(1 ,1)<… … <(r ,i ,(1 ,1)<(2 ,s ,t <(2 r ,3)<… …. ,3)<(2 ,2)<… … <( ,2)<(1 Let us put V, .= 帆J, . ,. (8 ). i Vi 1(” β)<( )}; t=▽i t-≦コ{V” β ,t then, for any i and t t turns out to be open and closed in XX Y , Vi set. V F 刃 Vi t for l≦i≦r ,. 9 ) (. も=1. ′ 5 lowsimmedi 8 9 Then itfol ); t ) )and ( rom ( a ely ,( ,f . . 刃 V F 刃 U~ , . 10 ( ). . Vi・V F の. (11) Next , 6xing x E L, let. Wheneveri≠j .. Ti( lx×1 }; x)= 刃{x×1 k亡 Ui 。k 。. 12 ( ) then we have. k “,. ・ 1. 刃(刃 Ti( x))コ L×Y.. 13 ( ). . ing a countable union, we may represent it by 12 ) be (. ′ 12 ) (. Ti( )=Zsdx ) x. t =l , where S ー x)=××lnkfor some n and k such that Si l ceS x)areopenandclosed も )こ Ui i i も( t( . Si. ions and thei in ××Y,their 6ni l open and te intersect te unions r 6ni r diHerences are al , thei closed in X X Y. De6 ne ‐. 14 ) (. V′ )=S, x x) , .( .( , V′ i =S ) ー{V′岬;(α ( ( ) - )}う x x i i t t ,β)<( ,t. ′ )t rns out to be o en losedin X×Y then, for any i and t u and c p t , Vi set. 15 ) (. マ ◎ = 召 マ“( x);. ′ then we have 12 14 15 ) )and( ) rom ( ,f ,( , ( 16 ). ! V 気x )= 刃 Ti. i=l. i=1. ).
(5) . Nobuo Kimur a and 17) ( Set. V′ ・V′ X)-の Wheneveri≠j i ) j( ,. V′ ); x ii書 γi(. 18 ) ( then we have. 19 ) (. ー V乞=LXY,. ) (20. V′ ・V′ i j= ≠ whenever iキi .. 工n fact( lowsf fore only( 19 16 fV′ 20 18 )fol rom ( )and( ) ) )need be proved, l ・V′ i jキの ,(13 ,there then. Vi ) f f V~( i ion o ・V5( th the de t )キのforsome x and y, in contradiction wi ni ) y x , ,. Let. Mi=Vi十 V′“. 21 ) (. ,. then. f{U 22 )( ) {Mdl≦i≦r ( }i s an open reanement o a il≦i≦r} i b i 2 t ) the order of{Mi l≦i ( s t a ≦r} l mos . 22 i l only prove( b f( )of( )be 22 ( ) a )o ng evident weshal .. f Mi l ・Mi ・Mi キ の then . 2 ′)(V + V′)(V 十V′)キの (Vi i i ,十Vi . i 2 2 i 8 8 ′ Hence (Vi V V V V V )+( i .i .i . iぜ i )十(Vi .V′い Vぉ)+(Vi .V′ .V′ ) i i , 2 , 3 , , . 2 8. )+(Vも・Vぜ V乍 +(Vも・V与 十(Vも.V彰・Vi ・Vi )十(Vも‐V1 ・V1)キの 3 2 3 2. But every summa f 11 20 t nd of the l )and( ) e ‐hand side is empty by( . Thus it reduces to. ion. Th i fore Proposition 4 is shown. radi b a cont ct s proves( )of(22) ,there. Combining propos i ion 3 and proposi ion 4 we have the f 1owi t t o1 ng proposltlon, Proposi tion 5 ぜ ひ〃 of 財沈みα鑑 sPのBZ SO解; , rたe dメ粥鋤s. 〆“”(ズ× Y)=ヱ Reference s ing d ime i 1 〕 N, Kimura: on thecover 1964 〔 tspace ) s on of produc s , Proc . Acad ,Jap . ,49 , ,267-271(. l: The p工oduc lspaceand a me 〔2〕 E, Mi i l l t ofa norma t t be norma chae th r c need no I . er , Bul . AI . Ma ,Soc . , 69 1963 ) . ,375‐376(. l h ion theory ince i 3 〕 W, Hurewiczand H, Wal 1941 〔 t t ) 1 ・ ・ a I ・: Dimens on Ma r s( e , Pr .se ,.
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