Nagata, J. (Jun-iti)

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Nagata, Jun-iti

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On the latest dimension theory

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STICHTING MATHEMATISCH CENTRUM 2e BOERHAAVESTRAAT 49 AMSTERDAM zw 1960 - 007 On the latest dimension theory Jun-iti Nagata 1960 On the latest dimension theory by Jun-iti Nagata Although dimension theory for separable metric s9accs was established about 30 years ago by Brouwer, Menger; Urysohn, Alexandroff and other mathematiciansJ it had been supposed to be very difficult to establish a dimension theory for non-separable cases. In about 1950, the sum-theoremJ which took a leading role in the old dimension theory, was extended to normal spaces by K. Morita 1 ) and othe~ mathematicians, Following that development, a satisfactory dimension the·ory was established for non-separable metric spaces by M. Kat~tov (1952) 2 ) and K. Morita (1954) 3). The main part of their theory consists of extensions of theorems that were well known for separable mei:;ric spaces 5 but \'IC must notice that those extensions are:: possible by virtue of th2 latest development in general topology and especial-ly in the theory of coverings. To begin with let us give three definitions of dimensions. Definition 1. (Menger-Urysohn 1 s weak inductive dimension) i ind dim¢= -1 ii if for any neighbourhood U ( p) of any point p of a topo-logical space H 2 there exists an neighbou1"'hood V (p) of p s a t is f yin g l) ;. V ( p ) c. TJ ( p ) j ind clim of the boundary of v(p) ~ n-1; then ind d im R ~ n . Definition 2. (Strong inductive dimension) i Ind dim¢= -1 :Li if for any c1osed sets F, G with Fr1 G = 0J there exists an opon set such that Uc. l:kc. ~-G ., Tnd dim of the boundary of ,_: ~ n-1 ~ then Ind dim R ~ n. Definition 3, (Lebesgue 1 s covering dimension) If for evepy finite • T)Cn cover•ing Vl of R there exists an open cover-ing 71' with½<?/!,. :x order¥~ n+1, then dim R ~ n. Def.2 and Def.3 are especially important for the latest dimension theory. Actually Def,2 = Def.3 for every metric space R' was proved by Katitov and Morita; but whether Def.1 == Def.2 for every metric space or not, is as yet a difficult open problem. -· •,J -· From now fo:cth we wilJ. concern ourselves only with metric spaces. The following principal theorems are extended from separable metric spaces to general metric spaces in Kat~tov-Morita rs thaory. Sum-theorem. Let R12 i=1,2 ... be closed subsets with dim R1 i n; then dim U R. s n. i=1 l -Decomposition-theorem. dim R~ n if and only if there exist n+1 O-n+1 dimensional subspaces R.J i=1 ... n+1 such that R - U R1 . l i=1 Product-theorem. dim R1 x. R2 i dim R1 + dim R2 . On the other hand, the following is Menger-Urysohn 1 s inbedding theorem whose extension to non-separable spaces was left open even in Kat~tov-Mori ta I s theory. Imbedding-theorem. A separable metric space R has dimension ~ n if and only if it is homeomorphic to a subset of the set H~n+1 of points in E2n+1 at most n of whose coordinates are rational. )I ) This problem was recently solved by the spealccc ·-r- by imbedding every metric space R with dim R i n into a subset 311 of the generalized Hil-bert space H, which was set up by C.H. Dowker. Although most of the abovementioned theorems are Just extensions of theorems that have been well known for separable metric spaces, there are also quite new types of theorems that had b3en unknown even for separable spaces and were established for general metric spaces quite recently. The speaker will devote the rest of his lectures to those nGW develop-ments of the theory. Fi1"st of all let us re cu 11 ;3c:i1:"te -:c the new test theorems for n-dimension-ality. K. Morita proved in 1954 S) the following theorem. 'rheorem 1. dJ.m R §. n if and only if there exists an open base Vl such co that 7/t == u U .. , vt. is a locally finite sys tern of open sets, order i='l :L - l { 1i - U I U ~·Vt_}~ n. It is very interesting to find an anaJ.ogy b1::tween this theorem and the metrizability theorem due to Yu. M. Smirnov and the speaker. The following test theorem due to the speaker in 1956 S) has also a remarkable analogy with the metrizability theorem of P, Alexandroff and P, Urysohn. ~Theorem 2. dim R il n if and only if there exists a sequence Vl., i=1 i 2 .. l of open coverings such that i Vl :>Vl* ~ '1 2 21'.,2 > V(.*3 > •• '., ii fs(p~Vti) I 1 == 1.,2 ••• J is a neighbourhood base of each point p of n. iii ordeI' ii-ii~ n+1 J i=1 J 2 ... W. Hurewicz and C.H. Dowker also proved in 1956 6) a similar theorem. Theorem 3. chm R:;; n if and only if there exists a sequence 1=1,2 ... such that m,, l i ii iii for ever•y Uc:. 7.Jl. 1 there exists V $ VC satisfying T.r c V :1 l-t- l = max. diameter { D \ li 6 u't1 } --,. O as i -• oo, order 11+1, We must not~ce the fact that these thcor2ms hav2 greatly simplified the test for n-dimensionality of metric sJacesJ because it suffices to show JUSt the existence of a s2qucnce of covering to prove then-dimensjonality of a metric space by virtue of these theorems. On the othe 1.' hand., if we J:--estr:Lct ourselves, for example~ to the definition of covering dimensionJ then we must show th2 existence of a refinc,ment ?Jf.. 1Jith order ?)t,,:, n+1 for every open covering Vl . To r.2vievJ :::rnr'S of the results deduced from these ne1/J test theorems, let us recall some definitions. Definition 1' ._ A space R 1s called countably dimensionnl v-1:, ,_-!-, ,_--,:::'.r:;,;,_:. t ta the Gecumposition theorem, if it is expressed as a union of count-ably many 0-dimensional subspaces. Definitio_n 5._ Let _o_ be a set. Let N(.D) =-= {(x1 :1x 2 , ... ) l xi <c. .0., 1~1,2 ... ·}· We define a metric f(x,y) for two points x=(x 1 :1x2 , ..• ) I ( ·) f' ·,r( <' ) b ( ) 1 cHlC. y=~ y '1 _; y 2 ~ • . . 0 h \ -•L y f X 3 y :::: -------min { i !x 1 lY.,} - .L Then N(..O.) rnakeiJ D 0-dimensional metric space an(kcalled a generalized t'"')~_ E7'° Da:u:-e 1 s 0-cLLn,ensional space. 'I'he spea l:er. :?ucceeded in '1958 7 ) by virtue of theor'em -1 to extend a chmensJou t. ·n•y from finite dimensional :11ctr•ic s·LJaccs to countably dimensional metric spaces, which had been an oJen problem till then even for separable metric spaces. The following is one of his prin-c :L r)a 1 :e(: c:;t1l tr:.: ~ Theorem 4. ~ metric space R is countably dDn2nsional if and only if there exi::.1to 9. E,ubset .S of N( Jl) for suitable .n and a closed con-tinuous mapping f of S onto R such that for ~ach point p of R, the inverse image r- 1 (p) consists of finitely many points. It is also an interesting problem how to characterize the dimension of a metric space by the metric which the space allows. This idea was realized, as for general metric spaces, first by J. de Groot and H. de Vries B) who proved the following theorem in 1955. Theorem 5. A metric space R is 0-dimcnsional if and only if it allows a non-APchir,1cdca11 rnetcic. 1~e speaker generalized this theorem by virtu~ of theorem 2J in 1956 S) as follovrn. Theorem 6. A metcic space R has dimension ~ n if and only if it allows a metrlc p(xJy) such that f,Jr every E. >0 and fop every point p of R, ) ,..,. ,, • .. C. ~ l i='I, , . n+-2 imply o(--, ,·1) --; ( . . 9 ,., j "·· t. for- some i.s j with i ,/ J; whcr'c st:.;2 (p) denotes the & 2 neighbourhood with the centsr at p. J. de Groot slrnpJ..i.:ficd this theorem in 'l95f3 9 ) as follov1s. Theorem 7, A sep2rable metric space R has dimension ~ n if and only if :'Lt allmr.Js r metric p(x;y) such that for 0vc:1"y e. >0 and for every point p of Rs 1 = 'l .. , n+2 imply p(q. ,o_ .. )< €. for some i,J vJith ifj. • 1. J Theorem 1 and theorem 2 or 3 arc showed to be useful for proving some types of imbedding theorems; too. Finally the spcak~r would like to inform of new progresses in the theory on d im2nsion and mapping VJhich a rie due to K. Nagami and J. Su-zuki 10 ). One of.their results is the following which refines the well-known theorem of U, Hurevncz. Theorem 8. Let f he a closed continuous mapping of a metric space R onto a metric :Jpacc ,S such that f-·1(q) consists of Just k points for every po1nt q of S; then dim R = dim S, 

On the latest dimension theory

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