階位空間におけるBaireカテゴリー定理について

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(1)Title. 階位空間におけるBaireカテゴリー定理について. Author(s). 桜田, 邦範. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 28(1) : 1-10. Issue Date. 1977-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6005. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Section II A) Vol. 28, No. 1 September 1977. -miaM^'lN5^ ( ^ 2 Â) ^ 28 ^ ^ 1 ^- BgW 52 ^9 ^. On the Baire Category Theorem in Ranked Spaces Kuninon SAKURADA Mathematics Laboratory, Sapporo College, Hokkaido University of Education Sapporo 064. IfS^HS^^H^ Baire ^ T =?" tj -ÎS^^v.-C. ^s^l. ^m^±îm/^?^^. Abstract Prof. K. Kunugi founded,in 1954, the method of ranked spaces, in the Note [ 3 ], as a generalization of that of metric spaces.. In the Note [ 3 ], for a complete ranked space in which the system of preneighborhoods satisfies the axioms (A),(B) and (C) of Hausdorff ([ 1 ]), he proved the following generalized Baire category theorem : Dans les espaces ranges complets, aucun ensemble ouvert non vide nest de premiere categone.. After that, H. Okano ([10]) succeeded in proving that the same theorem holds without the assumption of (C) by modifying some notions. Recently, S. Nakanishi ([ 9 ]) showed that Kunugi s result holds, without the assumptions of (B) and (C) by modifying the notion of nondense, for a complete ranked space of indicator ajo (coo is the first nonfinite ordinal number). In this paper, we study the Baire category theorem for a ranked space whose indicator is an. arbitrary inaccessible ordinal number. By the method used in [10],we can show a generalized Baire category theorem,which coincides with the result of Nakanishi ([ 9 ]) in the case' of a ranked space of indicator wo, and which is a generalization of the result of Okano ([10]). 1. Definition 1 ([3]). An ordinal number a is said to be inaccessible, if a is a limit number and if,for every /9 with /? < a and for every function /(y) defined for 7 with 0 ^ y < ,9, such that 0 ^ /(y) < a, we have always sup _/(y) < a. o&r<p. Proposition 1. Let a be an accessible limit ordinal number. Then there exists an inac-. cessible ordinal number a* such that 0 ^ a* < a, and there exists a mapping / defined for every /j. with 0 ^ p. < 0'*, such that 0 ^ /(î) < a, which satisfies the following two properties (1) and (2):. (D.

(3) K.SAKURADA. (1) a= _sup^f{^). Q£/"<-ff*' '. (2) If 0 ^ î<^2<ai!, then /(î) < ,(,^2). Proof. Throughout the proof of this proposition,we denote by W (r) the set of ordinal numbers p. such that 0 ^ H < r. Since a is an accessible limit ordinal number,we define a*. to be the smallest ordinal number y^W{a) such that there exists a mapping h of ^(y) into W{a) having the property that Q' = sup h{ p.). Let h be a mapping of W{a*} into W{a) such that a = sup h{ju). OSft<0!*. (I ) We first show that a* is a limit number. Suppose a* is an isolated ordinal number. Let a be the ordinal number such as o-+ 1 = a .. If the mapping h' of W{a} into W{a) is defined by putting /z/(/^) = /z(/^) for every H^ W{a), then sup h' {/j.} < a by the definition of a*. Since a is a limit number, there os/^^cf. is an ordinal number /? such that sup h' (//) < /? < a. For this y3 there exists an ordinal o^f^<cf. number ^06 W (a*) which satisfies that /? ^ /z (ô) < a, since a = sup h (^) . Since o^fi<a*. sup h{]u) = sup h' (/^) < /9, it follows that /ô = a. Hence it follows that sup h (//). OSft<ff. OSsfKff. .. .. OÂI^ff*. = sup ^h (n) = h{a} < a. This contradicts the definition of a*. OSft~<ff. (II) Next, we will show that there exists a mapping /of W{a*) into W (a) which has the required properties ( 1) and ( 2 ). First, we define a mapping g of W{a*} into ^(c?*) by transfinite induction in the following way: ^ ( 0 ) =0, g { 1 ) is the smallest ordinal number ij.^W{.a*) such that h(g{0)) <h{n). Let /9 be any ordinal number such as 0 < 0 < a*. Suppose that g (/^) has been defined for every p. € W{/3} and that h {g (0 )) < h{g( 1 )) <.••...< h ( 5' (^)) < •••• for every n^W{^}. If /3 is an isolated ordinal number, there is the ordinal number §• such that (5-+ 1 = y9. Since a is a limit number, and since a = sup h {u), we define q(Q} to Osafi<a*. be the smallest ordinal number /i^W{a*} such that h ( 9 (^)) < h (^). If ^ is a limit number, then the relation sup h (<7 {u )) < a. holds by the definition of a*. Since o' is a 0^ft'<P. limit number, and since a= sup h(/u), we define g{/3) as the smallest ordinal number os/jKa*. U^ W{a*} such that ÛP^^h {g (//))< /z (/^). From the construction it is evident that g (a) < g (/?) and h {g (a)) < h {g (/?)) for all a,j3with 0 ^ a</?< a*. Hence putting f=h^g, this shows that the mapping / of W{a*} into W(a) is strictly monotone increasing, i.e., /(î) < ,(^2) if î < U2 and ^, ,,2 € ^T(a*). Next, We will show that a = sup f(/u). Let /9 be any ordinal number such as Q ^ o^p-<a*. < a. Since a is a limit number, and since a =AUP...h (u), there is an ordinal number /ê OSfit<0!*"' "~~ ". W{ a*) so that /? < h (^/?) < a. On the other hand, the monotonicity of g implies H^.g^n) for every n e W{a*}. Suppose that ms<g{p.^ and h{g{/up))< h^ju/s) ; then we have. g {^+ 1) = min {^;/z(^(^/?))< /z (^), ^ e T7(a*)} ^ ^ from the definition of g ; and then,. (2).

(4) On the Baire Category Theorem in Ranked Spaces. as g{^}<g{p.pjr 1 ), we have g (^s) < ^, but this is a contradiction. Hence it follows that. h (^) ^ h{g {/up)). Therefore /?< /z(^) ^ /z(^(^/?))= /(^) < a. This proves that. " = AUP. o^fi<a* J v("-/'. (Ill) Finally, we will show that a is an inaccessible ordinal number. By (I), a* is a limit number. Suppose that a* is an accessible limit ordinal number. Then there exist a limit number a\ with 0 < a\ < a* and a mapping /i of W{a\) into pF(o'*) such that a* = sup f\ (^). Let /3 be any ordinal number such that 0 ^ /? < a. Since a = oê~<ai. sup f {/u) from (II), there is an ordinal number n^W{a*} such that f3 < /(^)<Q"and, for this H, there exists an ordinal number êW (,ai) such that n. </i (^), since a* = sup. o^f<a\. /i (^). Therefore, since the mapping / of W(a*) into ^(a;) is strictly monotone increasing,. it follows that /(^) < /(/i(^>). Hence /? < /(^) < /(/i(^)). So we have a = sup _ (/°/i)(^) for the mapping /°/i of W(ai) into 'PT(a), but this contradicts the definition. O^^<QI. of a*.. 2. Let us recall the notion of ranked spaces.. Definition of the depth of the space. Let R be a space endowed with such a structure that each point p of R has a non-empty family { V{p)} of subsets of R satisfying the axiom (A) of Hausdorff. V(p) is called a preneighborhood of p. We denote the family of all preneigh-. borhoods of p of R by ^S(p), and put %= { V(p) : V{p) e ^S(p),pe R}. A monotone decreasing sequence { Va {?)} of preneighborhoods of p^ R is said to be type 7, where y is an ordinal number, if a runs through the set 0 ^ a < 7 of all ordinal numbers and if. Va (P) ^ Vft {?) for all a,/9 with 0 ^ a </3 < r : (D Vo (p) ^ Vi (p) ^ ............ =? ^ {p) 3 .......... o ^ a< r.. If, moreover, there is no preneighborhood U(p) e %(j)), such that H Va {?) =? U{p}, the osa<7. sequence (1*) is said to be maximal. If a point p of R has no sequence of preneighborhoods which is maximal, the depth of R at p is said to be actually infinite. In another case, the depth of R at p is defined as the smallest ordinal number of types of maximal monotone decreasing sequences of preneighborhoods of p, and is denoted by a)(R,p). The smallest ordinal number of the depths w {R, p) of R at points p e R, is called the depth of the space R, and is denoted by co (R). Definition of the ranked space. Let wy be an inaccessible ordinal number such as 0)o ^ 0)v ^ CO {R~). If, for every ordinal number a such that 0 ^ a < a)v, a family lla of. preneighborhoods is newly endowed and the following axiom {a) is satisfied, then R is said to be a ranked space of indicator cov '.. (a) For every point p of R, every preneighborhood V(^) of p and every ordinal number a such as 0 ^ a < wv, there exist an ordinal number /9 and a preneighborhood U{p). of p such that a ^ /? < ^, ^7(^)g V(^) and ^/(^) e l^.. (3).

(5) K. SAKURADA. The members of [\y are called preneighborhoods of rank a. Definition of the fundamental sequence. Let R be a ranked space of indicator wu, and let (jDp. be an inaccessible ordinal number such as coo ^ ajfi ^ cdy. A sequence { Va {pa)}osa<^,. of preneighborhoods of points is called to be a> ^-fundamental, if the following four conditions (1), (2), (3) and (4) are satisfied: (D Vo{po) ^ V^pz)^ •••••• ^ Va(pa)^ •••••••••,0^ a< ^.. (2) Va{pa} € U,^ 0^ a< ^. (3) T-o ^ n ^ ••••••••• ^ra^ ••••••••• ,0^ ff< ^. ( 4) For each a such that 0 ^ a < <z>/x, there is an ordinal number /9 such that a ^ /9 <0)fi, Pft = Pft+^ and TP < r/?+i.. Definition 2 ([10]). A ranked space R of indicator co u is said to be complete if, for every oofi -fundamental sequence { Va {pa)}oâ<», , the following two conditions ( 5 ) and ( 6) are satis-. fied: ( 5 ) If w^ = wv, then ^ H V<, (^) ^ ^. O&CKw^. (6) If co^ < u^, then C} I {Va(pa)) ^ ^. osa<w^. On this, for a subset E of R, we denote by I{E) the set of all points p oi R such that there exists a preneighborhood V {?} of p which is contained in E. Definition 3 ([3],[9]). Let R be a ranked space of indicator cDy. A subset E of R is said to be nowhere dense in R if, for every point p of R and every preneighborhood V{p) of p, there exist a point q oi R and a preneighborhood U{q) of q such that U{q)t=V{p) and U{q) H E = <f). A subset F of R is said to be of the first category if it is a union of wvsequence of nowhere dense sets : F = U Ea where every Ea is nowhere dense. All other OÔKWfi. subsets of R are said to be of the second category. A subset E of R is said to be dense in R if, for every point p of R and every preneighborhood V{p) of p, V{p)C\ E ^ ^. 3. Lemma 1. Let R be a ranked space of indicator a)v. H ^ satisfies the axiom (B) of Hausdorff, and if Q<a<co(R), then, for every monotone decreasing sequence {V/s (pp)}o^/s<a of preneighborhoods,. _n j(y,(^))=/(_ny,(^)).. osp<a. osi3<a. Proof. It is clear that. .n /(y,(^)) ^/(.n y,(^)).. 0^ft<a ~ O^P<ct. On the other hand if p <= n KV^pp)}, then, for every ordinal number /9 with 0 ^ j3< a, Q^P<a. there exist a point p and a preneighborhood Uft(p) oi p such that U (p)^Vp{pff). We write Wo {?} = U°{p}. Let 7- be an ordinal number such that 0<7"<o'. Suppose that a sequence. {Wp {p}}o^p<r of preneighborhoods of p has been defined and it satisfies the following two conditions : Wo {?) 3 W, (?) ^.........3 Wp {p) ^........... Wft {pp) ^ Wp {?) for every 0 with 0 ^ /3<r.. (4).

(6) On the Baire Category Theorem in Ranked Spaces If f is an isolated number, then there exists the ordinal number / so that r + 1 = ;r, there -. fore,by axiom (B), there exists a preneighborhood Wy(p) of p such that Wr{p)c=W-y'{p)^\ Ur (p). If r is a limit number, then it follows from r<w{R) that the sequence {Wi3{p)}os/3<7 is not maximal, hence there exists a preneighborhood U\p) of p so that U'{p}'^ H Wfi(p). OSft<r. Therefore we can choose a preneighborhood W^p) of p such that Wr{p)^ U'(p)r\ Ur{p). Thus by the transfinite inductive process the sequence {W/s (,p)}os/3<a of preneighborhoods of p is defined. Since a < co (R), there exists a preneighborhood W(p) of /) such that W{p). ^ n w/s {p) ^ n Vp (p/s), therefore it follows that p e / ( H y^ (^^)). o^/2<ff. ~. o&p<a. ~. o^ft<a. Lemma 2. If R is a complete ranked space of indicator <z>y and if a is a limit number such as a < oov, then, for every monotone decreasing sequence {Vft(.p^}o^/3<a of preneigh-. borhoods which satisfies the following four properties (1), (2), (3) and (4), it follows that. ,n_j(y,(^))^ ^.. 0^^<ff. (1) Vo{po) ^ Fi(î) ^ ••••••••• ^ y,(^)3 •••••••••,0^/9< a. (2) T-O ^ 7i ^ ••••••••• ^rp^ ••••••••• ,0^/9< a.. (3) y^(^)eu^, 0^/?< a. ( 4) For every even number /9 such that 0 ^ /9 < a, p/s =p/3+i and ^ < ^+1. Proof. First, 9Uppose a is an inaccessible ordinal number. Since R is complete and { V/s. (p/3)}o^/3<a is an a-fundamental sequence, it is clear that H / ( V/s {p/s)) ^ ^. Next, suppog^<ct. ose a is an accessible ordinal number. Throughout the proof of this lemma, we denote by. W(,/u) the set of ordinal numbers A such that OÂ</u. By Proposition 1 there exists an inaccessible ordinal number a*,0<a*<a, and there exists a mapping / of W(a*) into W(a) such that a = sup _j{/jt) and that 0 ^ //i < /U2 < a* implies /Cî) < ,(^2). Now for 0^ p- â ^. every even number ^ 0 ^ p. < a , we define. /(/,)-1 if /(/,) is odd, f/(^) if /(^) is odd, g (^) = \' ; '. g ^ +1 )= f{.n} if /(//) is even, [ f (.ju) + 1 if /(/^) is even, where f{,u) — 1 means the ordinal number so that (/(^) — 1 ) + 1 = /(^). Then it is clear that g is a mapping of W{a*) into FF(a) such that S^P g(^) = a. ••••••••• ( ^ ) Sinece R is complete, for an a*-fundamental subsequence {Vg(ii) {pg(^))}os^<a*, of {Vp(p/s) }o^p<a, it follows that H / ( Vg(fi) {pg(/^))) ^ ^. For every ordinal number/?, 0 ^ ^ < Q', 0;S,u<ff*. by (^), there exists an ordinal number /u such that0^<o;* and /3<g{/u)< a, and therefore Vp (^) ^ y^^) (^(^). Hence _ Q / ( V/, (^)) 3 _ H . / ( Y^/.) (^(/.))). On the other o^ft<a ~ o^ft<a*. hand, since {Vgw {pg(ii) )}os^<a* is a subsequence of {Vp (^)}o^<a, it is clear that Ft. o^ft<a*. 7(F^)(A^)))=3 _n 7 (V, (As)). Hence H / ( F, (^)) = _n^, / (Vgw (^(/.))) osp<a. .. ~. osft<a. o&f^<a*. ^ ^. Theorem. If R is a complete ranked space of indicator o)v, and if % satisfies the axiom. (B) of Hausdorff, then, for any subset A of R of the first category, the complement of A is. (5).

(7) K. SAKURADA. dense in R. Proof. Since a set A is of the first category, without loss of generality we may suppose that A = U E 2 a, where each Eza is nowhere dense in R. 0^ff<Uy. Let p be any point of R, and let V{p) be any preneighborhood of p. Since Eo is nowhere dense in R, owing to the axiom {a), there exist a point po, a rank 7-0 and a preneighbor-. hood Vo(po) of po such that Vo{po)^ V(p), Vo{po)HEo = 0 and ^(AÛro. For this Vo (.po), by the axiom {a) there exist a point pi, a rank 7-1 and a preneighborhood Vi {pi) of pi such that Vi(pi)^Vo{po), p\=po, ro< 7-1 and 1/1 (î)6U,.,. Let a be an ordinal number such that 0<o'< cov. Suppose that we have already defined p^, ^ and V/s {p/s) for all ,9 such that 0^/9< a, and they satisfy the following conditions: Vo (po) 3 V, (^) ^ ......... 2 y, (^) ^ ......... ,. ,^(AÛ. ro ^ n ^ ......... ^ ^. and, for every even number (S, p/s = pp+\, r^ < 7-^+1 and Vp {pp} n Efi = 0. i ) If a is an even and isolated ordinal number, then, for the ordinal number S' such as §• + 1. = a, it follows from the fact that Ea is nowhere dense in R, and by the axiom (.a), that there exist a point pa, a rank ra and a preneighborhood Va(pa) of ^ such that Va{pa)^Vs{ps), Va {pa} H Ea = 4>, Va {?a) ^ U ^ and Ts < Ta. ii) If a is a limit number, then, since R is complete and a < co v, Lemma 2 implies that there exists a point q such that q <= Hy^ (j^/s). On the other hand, since a < co (R), Lemma 1 o^lKa. shows that H / {V/s (p/s)) = I { D Vp (^)). Hence there exists a preneighborhood U(q) o^P<a. o^j8<ff. of q such that U{q) <= Fl V^(^^). Since £'a is nowhere dense in 7?, by the axiom {a) there o^P<a. exist a point pa, a rank 7^ and a preneighborhood Va {pa} of ^ such that y^ (â) ^ U(q), Va {Pa) n Ea = cf>, Va {pa} Û ^ and SU?_ Tft < Ta. o^^<a. iii) If a is an odd number, then, for the ordinal number 5 such as 5~+ 1 = a, putting ^ = ps, by the axiom (<z) there exist a rank ra and a preneighborhood T^(â) of j)a such that Va{pa) ^ Vs {ps), rs < ra and Va {pa)€ U ^. Thus we obtain an (ziy-fundamental sequence {Va (.pa)}oâ<^, such that ( H Va {pa}} vv>. 'osa<^. n ( _ U Eza} =0 and n VaW ^ V{p}. Hence we have ( n Va{pa}} H A = 0â'<(dy. Oiaff<(yy. '^. '. 0^0'<(uy. Since R is complete, there exists a point q* such that q* e n yâ). Therefore we oâ<^. have q*e V{p) H (R\A). This shows that R\ A is dense in R . 4. Definition 4. Let 7? be a ranked space of indicator a)v. A set E ^ R is said to be oên if, for every point p of E, there exists a preneighborhood V(,p) of ^ such that V{p}^ E. A set F <s= R is said to be closed if its complement R \ F is open. Moreover, for %, we consider the following condition ( C*) : (C*) For every point p of R and every preneighborhood V{p} of ^, there exists a pre-. (6).

(8) On the Baire Category Theorem in Ranked Spaces. neighborhood W{p) of p such that W(p)^V{p) and such that, for every g^W(p), there exists a preneighborhood U{q} of q such that U{q) ^ V{p). The proof of the following is clear, and will be omitted. Proposition 2. Let R be a ranked space of indicator iziy. % satisfies the condition (C*) if and only if, for every subset A of R, I {I (A)) = / (A). Proposition 3. For a ranked space R of indicator o)u, let us consider the following ; ( a) For every subset A oi R which is of the first category, its complement R\ A is dense in R.. (/9) Every non-empty open set in R is not of the first category.. Then, we have : ( i ) (a) implies (/9); (ii) If % satisfies (C*), then (/?) implies ( a), Proof. ( i) Suppose that there exists a non-empty open set GC=sR of the first category.. Then, by (o), its complement R\G is dense in R hence I {G} = (p. Since G is open, this shows that G = I {G) = (f>, but this is a contradiction. (ii) Suppose that there exists a subset A ^ R which is of the first category and its. complement R\A is not dense in R. Then 7 (A) ^ (f). Since SS satisfies (C*), I {I (A)) = I (A). This means that /(A) is a non-empty open set in R. Therefore, by (/?), Z (A) is not of the first category. Since A =i> 7 (A), A is not of the first category, but this is a con tradiction. Remarks 1) We note that, in the case where R is a ranked space of indicator wo, Lemma 1 is not necessary for the proof of the previous theorem ; namely, it is not necessary that the. system of preneighborhoods in R satisfies the axiom (B) of Hausdorff. Hence, in this special case, the previous theorem coincides with a result of Nakanishi ([ 9 ]). 2) By Proposition 3, the previ :>us theorem is a generalization of a result of Okano ([10]). 5. Example. Let us give an example of ranked space which has not the property of (a), but has that of (/?) in Proposition 3. (I ) Let R be the set of all real numbers. For each point p of R, we define preneighborhoods of p as follows : Let n and m be ordinal numbers such that Q^n,m<wo. Then, if p -=^ 1 — |L i , we m + 1 '. define V{p; n} = [q;q^R, \p - q\ < ^ ^ }, and if p= 1 - ^ ^ , we define V{p; n) = [q;q^R\p-q\ < ^ ^ } and V* {?) = {q;?^ q < 1 and ^ is rational}. We denote the family of all preneighborhoods of pe. R by %(^), and put ^ = U {^S(.p) ', p £ R}. Then % satisfies the axiom (A) of Hausdorff, but not the axiom (B) of Hausdorff and the condition (C*). And it is clear that the depth of R is coo. Therefore, for every n, 0 ^ n < a)o, we define a family Un of preneighborhoods of rank n as follows :. l\n={V(p;n); p^R} U {F*(^);^=l--,J:r-,,0^ m< ô}. Then it is easily verified that R is a ranked space of indicator a)o, which is not complete.. (7).

(9) K.SAKURADA. (II) Let us first show that, for every point p of R, the set {?} is nowhere dense in R. Let pc R be given. Let q be any point of R, and let U{q} be any preneighborhood of q. We shall define a point r of R and a preneighborhood U{r) of r as follows : Case 1: If U(q) $ p, then we define r and U{r} by r = q and ^7(r) = [/(<2'). Case 2: If U{q}^p and U{q} takes the form of V(q ; n), and if ^ = g, then we. define r and U{r} by r = p +-2-[^-Y) if P^ °> ^ = ^ - 3 (^+ ^ ) if p <0, and U{r) = V(r ; 4n+ 3 ). Caes 3: If U{q} 3p and U{q) takes the form of V{q; n), and if p ^- q, then we choose an ordinal number wo such that ^ —'-.<^ \p-., — q\ and 1 ^ mo < coo, and - — -.-" - —,. define ,r and (7(r) by r= ^ and ^/(r) = V{r ; 2mo). Case 4: If U{q}3p and U(g)=V*{g), then there is an ordinal number no such that 1 ————o and 0 ^ no < coo, and we define r and U(r) by r = 1 %0. -]—^ and U{r} = V* (r). no +2. Then it is clear that U{r)^ U{q} and U{r)n{p} = 0. Therefore {?} is nowhere dense in R.. (HI) We next show that R has not the property of (a) in Proposition 3. Let us consider the set Q of all rational numbers. By (II), Q is a set of the first category. But its complement R\Q is not dense in R, since V* (1 — ^J~\ -, ) Fl (R\Q) = <^, 0 ^ n ^ _(__[_/•• v-- I -C. / Y-,. < coo.. (IV) Finally, we show that R has the property of (^) in Proposition 3. First let us show that every non-empty open set G of R contains always a set of the form. {a,b)= {p; p^ R, a < p < b}, where a, b^ R and a < b. As. G ^ (f), there is a point p of G. Since G is open, there exists a preneighborhood ?7(^). of p such that U(p)^G. If U{p) takes the form of V(p ; n), then it is clear that {p——1—,, ^+—1—, ) =V{p; n)=U(p) ^ G. If U{p)= V*{p}, then there is an ordinal number no such that 0^%o<^>o and p = 1 — _ ", -, ; hence, for a rational number. q such asp < q < 1 — _ J~, ^ , the preneighborhoods of q of R take only the form of V{ q ; n) ; but, since q^V*{p)= U{p) ^ G and G is open in R, there exists an ordinal number m such that 0^%i<ô and V(.q ; n\) S G ; therefore we have {q———^ , q+——i ). = V{g; m)^ G. Consequently, in order to prove that G is not a set of the first category, it is sufficient to show that each set of the form (a, b~) S= R is not of the first category. Moreover, without loss of generality, we can assume that {a, b) F\ { 1 — ,, \ i ; 0 ^ n < wo} = (j).. (8).

(10) On the Baire Category Theorem in Ranked Spaces Let E be an arbitrary set of the first category, and let E = U Ezn where every 0^n< wo. E \n is nowhere dense in R. For a subset A of 7?, we denote by A the closure of A with respect to the usual topology on R. For every pe (a, b), we choose a preneighborhood V(pn) of p such that (a, b)SV{p; n). Since Eo is nowhere dense in R, for this y(^); n), by the axiom {a) of ranked space we can choose a preneighborhood V(po ', no) of po such that 0 < no, V{p; n) ^ V{po; no) and V{po; no)F\Eo = 0. Hence putting pi = po and n\ = 2no, we have V{po', no) =) V{pi ; n\) 3 V(î ; Hi). Let ^ be an ordinal number such that 1 < k < (DO. Suppose that, for all ordinal numbers i such as 0 ^ i < k, we have already defined pi, Hi and V(pz; n,), and they satisfy the following condition ( 1) and(2): d) V(A>; no) ^ y(î; î) 3 ••••••••• 3 y(^.; n,) ^ •••••••-, no ^ n\ ^ ......... ^ ^^ ......... ^. ( 2 ) For every even number i, pi = p,+i, m+x = 2n; and V {pi; ni) nEi = (j). If k is an even number, then, since E k is nowhere dense in R, there exist a point pk, an or -. dinal number nu and a preneighborhood V{pk', nk) of pk such that y(/)A-i ; HA-O ^ y(^ft ; nk), nk-i < nk and V(,pk ', nk) Ft Ek = (f). ^ k is an odd number, then, putting pk = pk-i. and nk = 2nk-i, we have V(pk-i; Hk-v} ^ V{pk; nk) ^ V(pk ; nk). Thus we have a fundamental sequence { V {pi ', nt)}o^z<uo °f preneighborhoods. On the other hand, the sequence { V{pi; ni)}ost<ô is a monotone decreasing sequence of bounded. closed intervals with ^respect to the usual topology on R. Therefore there exists a point q of R such that _ D V (^z+i ; n2z+i) = {q}. Hence q e _H V (^zz+i ; ^21+1) ^ _ n y(^)f OSKaio. '. OSKwo. OS.KOI. "0. ; %,) g (^,^) . On the other hand, since ( H V{pi; ni) ) H ( U Ezi) = ^ we have. ,. ..,,. _. ,^,^/. .. .. '0£z<(u0. •". •. '. -•. •. '03az<u0. q^ U Ezi = E. Consequently, {a, b) is not a set of the first category. Odz"<nj0. Acknowledgement The author would like to express his hearty thanks to Professor K. Suzuki, Science Universityof Tokyo, for his valuable suggestions and warm encouragement. References [ 1 ] Hausdorff, F. (1914): Grundzuge der Mengenlehre. Leipzig, p. 213.. [ 2 ] Kunugi, K. (1951): Kaisekigaku Yoron. Tokyo, (in Japanese). [3] Kunugi, K. (1954) : Sur les espaces complets et regulierement complets. I. Proc. Japan Acad., Vol.30, 553556.. [ 4 ] Kunugi, K. (1959): Sur une generalisation de 1' integrale. Fundamental and Applied Aspects of Math. (published by Res. Inst. of Applied Electricity, Hokkaido Univ.), 1-30. [ 5 ] Kunugi, K. (1966): Sur la methode des espaces ranges. I, II. Proc. Japan Acad, Vol. 42, 318-322, 549-554. [ 6 ] Kunugi, K. (1969): On the method of ranked spaces. Noda Mathematical Pamphlet Series. 1. 1-15, (in Japanese). [ 7 ] Nakanishi, S. (1968): On generalized integrals. I. Proc. Japan Acad., Vol. 44, 133-138. [ 8 ] Nakanishi, S. (1974): On the strict union of ranked metric spaces. Proc. Japan Acad., Vol. 50, 603-607. [ 9 ] Nakanishi, S. (1975): The Baire category theorem in ranked spaces, Proc. Japan Acad., Vol. 51, 411-414. [10] Okano, H. (1957): Some operations on the ranked spaces. I. Proc. Japan Acad., Vol. 33, 172-176.. (9).

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