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Unifying the Baire category theorem Huber, George Michael 1970

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UNIFYING THE BAIRE CATEGORY THEOREM by ' GEORGE MICHAEL HUBER A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF-BRITISH COLUMBIA August, 1970 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission for e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e rmission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date jQJUJUYnlhvv | £, /97Q i ABSTRACT The f o r m u l a t i o n of the B a i r e category theorem found' i n most elementary topology te x t s deals w i t h two d i s t i n c t c l a s s e s of spaces:, l o c a l l y compact spaces, and complete metric spaces. This "dual theorem" status of Ba i r e ' s theorem suggests the problem of f i n d i n g one c l a s s of t o p o l o g i c a l spaces f o r which the B a i r e category theorem can be proved and which i n c l u d e s both the l o c a l l y compact spaces and the complete metric spaces. This t h e s i s surveys and compares the three approaches to t h i s problem taken by three methamticians-. The c l a s s i c a l r e s u l t s of E. Cech achieve a u n i f i e d B a i r e theorem by a Aefinl.ti.on of completeness d i f f e r e n t from that i n current common usage. Johannes de Groot introduced a n o t i o n of subcompactness, g e n e r a l i z i n g compactness. K. Kunugi worked i n the s e t t i n g of complete ranked spaces which g e n e r a l i z e uniform spaces and e l i m i n a t e the need to assume r e g u l a r s e p a r a t i o n i n the space. This l a s t p o i n t i s the b a s i s f o r the c o n s t r u c t i o n of a complete ranked space which i s n e i t h e r subcompact nor complete i n the sense of Cech. I t i s a l s o shown i n the paper that there e x i s t spaces subcompact but not complete i n the v sense of Cech, and that i n c e r t a i n s p e c i a l cases completeness V i n the sense of Cech i m p l i e s subcompactness. i i T a b l e o f Contents I n t r o d u c t i o n - -Page i i l T e xt v..':^ ' - Page 1 R e f e r e n c e s - • Page' 30 . UNIFYING THE BAIRE CATEGORY THEOREM Formulations of the B a i r e category theorem found i n most elementary topology t e x t s are a c t u a l l y two some-what d i s t i n c t theorems: A l o c a l l y compact Hausdorff space cannot be w r i t t e n as the countable union of nowhere dense subsets ( i . e . i s of second B a i r e c a t e g o r y ) ; a complete m e t r i c space i s of second B a i r e category. This p e c u l i a r s i t u a t i o n suggests the problem of f i n d i n g one p r o p e r t y of t o p o l o g i c a l spaces f o r which the f o l l o w i n g three a s s e r t i o n s are t r u e : Every l o c a l l y compact Hausdorff space enjoys t h i s p r o p e r t y ; every complete m e t r i c space enjoys t h i s p r o p e r t y ; every space e n j o y i n g t h i s p r o p e r t y i s of second B a i r e category. T h i s t h e s i s i s a survey of three papers by three mathematicians, each w i t h a d i f f e r e n t s o l u t i o n f o r t h i s problem. S e c t i o n 1 deals w i t h the c l a s s i c a l r e s u l t s of E. Cech [ l ] who proved the B a i r e c a t e g o r y theorem u s i n g a d e f i n i t i o n of completeness which i s more g e n e r a l than that c u r r e n t l y i n use. For m e t r i c spaces completeness i n the sense of Cech i s e q u i v a l e n t to t o p o l o g i c a l completeness. I t b e i n g the case that every l o c a l l y compact Hausdorff space i s complete c i n the sense of Cech, the d e s i r e d u n i t y i s achieved. In 1 9 o 3 Johannes de Groot - .apparently without r e f e r e n c e to the xwork of Cech - approached the u n i f y i n g problem w i t h h i s g e n e r a l i z a t i o n of compactness: •subcompactness. iv De Groot also had the intention of achieving a formulation of Baire's theorem which allowed the countability conditions to be changed to an arbitrary cardinality. In our presentation of his work in section 2 we have supressed these generalizations somewhat. Specializing to the standard classical definitions has facilitated the comparison of de Groot's subcompactness with completeness in the sense of Cech. This comparison is the content of the third section. Kinjiro Kunugi proves the Baire category theorem in the settings of ranked spaces which is a generalization of uniform spaces [5]. This work is of particular interest in virtue of being the only formulation of the Baire theorem known to us which eliminates a l l separability requirements. This important weakening of the hypotheses allows us to close section 4 with an example of a "complete ranked" topological space which is neither complete in the sense of Cech nor ' subcompact as defined by de Groot. r V ACKNOWLEDGEMENT The author wishes to acknowledge h i s indebtedness to Dr. J.V. Whittaker f o r h i s p a t i e n t guidance i n f o r m u l a t i n g t h i s t h e s i s . The f i n a n c i a l support of the N a t i o n a l Resear'ch C o u n c i l of Canada d u r i n g the' w r i t i n g of t h i s t h e s i s i s a l s o g r a t e f u l l y acknowledged. Throughout t h i s paper the d e f i n i t i o n s used w i l l f o l l o w the ustyage of R. E n g e l k i n g where h i s t e x t [2] c o n t a i n s the c o r r e s p o n d i n g terms. With important e x c e p t i o n of paragraph 4 the t o p o l o g i c a l spaces under c o n s i d e r a t i o n w i l l be r e g u l a r (T-^ s e p a r a t i o n ) u n l e s s a s t r o n g e r s e p a r a t i o n axiom i s s p e c i f i e d . I t may be h e l p f u l to r e c a l l (or make, as the case may be) these few d e f i n i t i o n s : Given two subsets U and V of some set X the s e t t h e o r e t i c d i f f e r e n c e of U and V w i l l be w r i t t e n as U\V; U\V i s the i n t e r s e c t i o n of the subset U wit h the complement ( i n X) of the subset V . We w i l l tend to use c a p i t a l Roman l e t t e r s from the f i r s t p a r t of the alphabet to denote f a m i l i e s of subsets, and to use c a p i t a l Roman l e t t e r s from the l a s t p a r t of the alphabet tc denote s i n g l e subsets. In keeping w i t h t h i s tendency one w i l l see the n o t a t i o n A = s e S where A w i l l be a f a m i l y of subsets U indexed by some s u i t a b l y l a r g e index set S . A f a m i l y of subsets i s s a i d to have the f i n i t e  i n t e r s e c t i o n p r o p e r t y i f each f i n i t e s u b c o l l e c t i o n of sub-se t s from the f a m i l y has nonvoid i n t e r s e c t i o n . A f a m i l y of subsets i s s a i d to have the descending c h a i n c o n d i t i o n i f every s t r i c t l y d e c r e a s i n g sequence.of subsets from the f a m i l y i s f i n i t e . A t o p o l o g i c a l space i s a B a i r e space i f i t s a t i s f i e s e i t h e r of the two e q u i v a l e n t c o n d i t i o n s (a) The i n t e r s e c t i o n of every countable c o l l e c t i o n o f open everywhere dense subsets -1-i s everywhere dense,, (b) The union of every countable c o l l e c t i o n of nowhere dense subsets i s a boundary set. A boundary  set i s a subset having v o i d i n t e r i o r ; a set i s nowhere dense i f f ( i f and only i f ) i t s c l o s u r e i s a boundary s e t . (1) COMPLETENESS IN THE SENSE OF CECH. This s e c t i o n i s devoted to a summary of the c l a s s i c a l treatment of the B a i r e Theorem of E. Cech who formulated the d e f i n i t i o n of completeness i n the sense of Cech i n 1957 [ l ] . Further b i b l i o g r a p h i c notes can be found i n [2] whose text we are summarizing, pages 142 to 146. Given a Tychonoff t o p o l o g i c a l space X , we s h a l l use a symbol of the form cX to denote a c o m p a c t i f i c a t i o n of X ; cX denotes a compact t o p o l o g i c a l space and c denotes a homeomorphic embedding of X i n t o cX . c(X) s h a l l denote the image of X under the mapping c , hence c(X) = cX .. I t i s w e l l known that every t o p o l o g i c a l space has a c o m p a c t i f i c a t i o n i f f i t i s a Tychonoff space. The symbol BX s h a l l be reserved to denote the Cech-Stone / c o m p a c t i f i c a t i o n . The f o l l o w i n g theorem i s fundamental to the d e f i n i t i o n of completeness i n the sense of Cech. THEOREM (1.1): Let X be a Tychonoff space ( T ^ s e p a r a t i o n ) ; then the f o l l o w i n g are equivalent. ( i ) For every c o m p a c t i f i c a t i o n ' cX of the space X the remainder cX\c(X) i s an F -set i n cX . a ( i i ) The remainder B A 8 ( X ) i s an F -set i n BX . ( i i i ) For some c o m p a c t i f i c a t i o n cX of X the -2-remainder cX\c(X) i s an F^-set i n • cX . A Tychonoff t o p o l o g i c a l space i s s a i d to be complete  i n the sense of Cech~i"f"~'i~t" s a t i s f i e s one of the equivalent con-d i t i o n s of Theorem ( l . l ) . An i n t r i n s i c c h a r a c t e r i z a t i o n of completeness i n the sense of Cech i s given by THEOREM (1.2): A Tychonoff space i s complete i n the sense of Cech i f f there e x i s t s a countable f a m i l y {A. I 0 0 of open 1 i = l coverings of the space X s a t i s f y i n g the c o n d i t i o n : I f • 0 i s a f a m i l y of closed subsets of X w i t h S S £ o (a) the f i n i t e i n t e r s e c t i o n property and (b) f o r each i = 1,2,... there e x i s t a v s (•__") a n d a n 0P e n subset IJ(i) e A^ w i t h v s c ^ ( i ) ' ^hen the i n e q u a l i t y n V £ 0 holds. To i l l u s t r a t e the r o l e of the f a m i l y seS s of open coverings, ^ i ^ i - i w e i n t e r p o l a t e the f o l l o w i n g example. Let X be the h a l f open i n t e r v a l (0,1] w i t h i t s usual topology. Then X i s complete i n the sense of Cech CO / where f o r the f a m i l y we may choose the s i n g l e open covering k = { ( l / n , l ] : n = 2,3,...l . Indeed i f [V "I i s a f a m i l y of closed subsets s a t i s f y i n g (a) and (b) of Theorem (1.2), l e t ^^^) ^he closed set'and the i n t e g e r w i t h V ^ - ^ c ( l / r ^ , . 1 ] . Then {V g n v " ^ ^ ' e g i s a f a m i l y of subsets of the compact t o p o l o g i c a l space [ l / ( n , + 1 ) , l ] . But t h i s new f a m i l y of cl o s e d subsets - 3 -r e t a i n s property (a) of Theorem (1.2), hence 0 <Vs n V s ( i ) ) ^ 0 ' seS s s^- ; We conclude that and a l s o that X Is indeed complete i n the sense of Cech. The f o l l o w i n g theorem summarizes some of the p r o p e r t i e s of completeness i n the sense of Cech. THEOREM ( 1 . 3 ) : ( i ) Completeness i n the sense of Cech i s a h e r e d i t a r y property w i t h respect to closed subsets. ( i i ) Completeness i n the sense of Cech i s a h e r e d i t a r y property w i t h respect to subspaces which are Gg-sets. ( i i i ) The t o p o l o g i c a l sum of a f a m i l y of d i s j o i n t t o p o l o g i c a l spaces i s complete i n the sense of Cech i f f each component of the sum is,. ( i v ) The C a r t e s i a n product of a countable number of t o p o l o g i c a l spaces complete i n the sense of Cech i s complete i n the.sense of Cech. Before e x p l i c i t e l y r e c o r d i n g Cech's form of the B a i r e theorem, two remarks are i n order. A l o c a l l y compact Hausdorff space v i s complete i n the sense of Cech. To apply the d e f i n i t i o n we s h a l l use c o n d i t i o n ( i i i ) of theorem C-( l . l ) . Indeed., the remainder of the space i n i t s Ale x a n d r o f f one p o i n t c o m p a c t i f i c a t i o n i s a s i n g l e p o i n t ; hence t h i s - 4 -remainder i s c l o s e d , so i n p a r t i c u l a r i t i s an F - s e t . In a the paper p r e v i o u s l y c i t e d [ l ] E. Cech showed that i n a metric t o p o l o g i c a l space the notions of topological:completeness and completeness i n the sense of Cech are eq u i v a l e n t . THEOREM ( 1 . 4 ) : (Baire) Every space completein the sense of Cech i s a B a i r e space. (2) SUBCOMPACTNESS OF DE GROOT The work of Cech gives a u n i f i e d B a i r e category theorem i n the sense that i t t r e a t s one c l a s s of t o p o l o g i c a l spaces which i n c l u d e s simultaneously l o c a l l y compact Hausdorff spaces as w e l l as complete metric spaces. J . de Groot [ j 3 ] deals w i t h another c l a s s of t o p o l o g i c a l spaces-subcompact ones-which a l s o has t h i s u n i t y . The work of de Groot a l s o explores the p o s s i b i l i t y of a fo r m u l a t i o n of Bai r e ' s theorem which replaces the c o u n t a b i l i t y c o n d i t i o n s w i t h m-conditions, m being any c a r d i n a l . A l l spaces are assumed to be r e g u l a r (T^ s e p a r a t i o n ) . v Let A denote a base of nonvoid open subsets of the space X . A nonvoid subset F of - A i s a r e g u l a r / f i l t e r base r e l a t i v e to A i f ( i ) Each set U e F contains some set U' e F w i t h U*' c U- , ( i i ) F has the f i n i t e i n t e r s e c t i o n property. Let F ="{U s] g s be a r e g u l a r f i l t e r base i n X r e l a t i v e to some open base. F i s preconvergent i f H U0• 0 . A t o p o l o g i c a l space i s subcompact i f there scS s - 5 -e x i s t s an open base A such that every r e g u l a r f i l t e r base r e l a t i v e to A i s preconvergent. S p e c i a l i z i n g to the countable case, a t o p o l o g i c a l space i s countably subcompact i f there e x i s t s some open base A such that every countable r e g u l a r f i l t e r base w i t h respect to , A i s preconvergent. Completeness i n the sense of Cech i s i n general only preserved by the f o r -mation of countable C a r t e s i a n products. Subcompact spaces enjoy the stronger property: THEOREM (2 . 1 ) : ( i ) The C a r t e s i a n product of subcompact spaces i s subcompact. ( i i ) The t o p o l o g i c a l sum of d i s j o i n t sub-compact spaces i s subcompact. Proof of ( i ) : Let ' [X 1 a be a c o l l e c t i o n of t o p o l o g i c a l — — — — — — — — — — S S £ o spaces w i t h X g subcompact w i t h respect toopen base A g f o r each s e S. I f the whole space X i s not an element s of A e f o r any s , then add X to the open base A so that ( A J p Q may be used to con s t r u c t a b a s i s f o r the S S 6 o Tychonoff topology of P X . I t remains to show that seS s p X i s subcompact wi t h respect to t h i s open b a s i s . seS s I f } -n i s a r e g u l a r f i l t e r base i n P X , x i fc x\ ~ S seS then f o r each s , the p r o j e c t i o n s {ir F } p w i l l be a o X X t Ix subset of A and a r e g u l a r f i l t e r base. But each X i s S -yt S assumed subcompact so D (TT- F ) / 0 f o r each s e S . ' reR ' s r Hence n F ^ 0 and P X i s subc'ompact w i t h respect reR r seS s -y to the base generated by the bases A s Proof of ( i i ) : Suppose again that {X 1 _ a , i s a c o l l e c t i o n S S 6 o -6-of d i s j o i n t t o p o l o g i c a l spaces w i t h X subcompact wi t h respect to open base A f o r each s e S . Let {F^}^ „ be a o x X t i \ r e g u l a r f i l t e r base i n the t o p o l o g i c a l sum of the X o and s e l e c t any r' from R . F ' i s then a union of b a s i s elements s e l e c t e d from among the bases A„ , s e S . Some of the base s sets of t h i s union may be the v o i d s e t , but not a l l of them; l e t X„, denote one of the spaces such that F , O X , ^ 0 . S X* s R e l a t i v i z i n g now to the subcompact space X 1 we can see that s {F^, fl X ,1 p w i l l be a r e g u l a r f i l t e r base i n X , . So in v o k i n g the d e f i n i t i o n of subcompactness, n (F 0 X ,) ^  0 . R x o A f o r t i o r i D F ^ 0 , and the t o p o l o g i c a l sum of the spaces reR r X„ i s subcompact. I f one imposes the r e s t r i c t i o n that the c a r d i n a l i t y of the index set R be that of the n a t u r a l numbers, the proofs j u s t given become a p p l i c a b l e to the COROLLARY ( 2 . 2 ) : ( i ) The C a r t e s i a n product of countably subcompact spaces i s countably subcompact. ( i i ) The t o p o l o g i c a l sum of countably subcompact spaces i s countably subcompact. Before proceeding to the B a i r e theorem f o r sub-compact spaces, i t i s i n order to present the f o l l o w i n g two theorems. THEOREM ( 2 . 3 ) : A l o c a l l y compact space i s subcompact. Proof: For one v having i n mind the Bourbaki d e f i n i t i o n of compactness, i t i s c l e a r that a compact t o p o l o g i c a l space i s subcompact r e l a t i v e to any open b a s i s . The l o c a l l y compact case i s reduced to the compact case as f o l l o w s . - 7 -Por each p o i n t x i n the l o c a l l y compact space X , s e l e c t some open neighborhood of x w i t h compact c l o s u r e , V . Let A be the f a m i l y of a l l open neighborhoods of x which are contained i n t h i s V . B = {A : x e X l i s x x an open b a s i s f o r the space X . .Suppose that P i s a r e g u l a r f i l t e r base i n B and s e l e c t one subset U from the c o l l e c t i o n P . Since F i s a s u b c o l l e c t i o n of B and each b a s i s element i n B has a compact c l o s u r e , U i s compact. As was noted at the onset the compact space U w i l l be subcompact. But the f a m i l y E = [U n W : W e F} i s a r e g u l a r f i l t e r base i n the sub-compact space U . Hence 0 ^ n ( u n w ) c n w WeF WeF which shows the l o c a l l y compact space X to be subcompact. The countable case r e q u i r e s no f u r t h e r proof. COROLLARY (2.4): A l o c a l l y count a b l y compact space i s countably -> subcompact. THEOREM (2.5): A m e t r i z a b l e space i s complete i f f i t i s subcompact. Proof: We s h a l l f i r s t assume that the m e t r i z a b l e space X i s complete and co n s t r u c t a base f o r the space w i t h respect to which X i s subcompact. We w i l l s e l e c t a base from among the open sets of diameters 1/n, n = 1,2,..., by means 1 of the f o l l o w i n g A -8-LEMMA: Every cover of a set by a c o l l e c t i o n of subsets has a subcover s a t i s f y i n g the descending chain c o n d i t i o n . Proof of lemma: The lemma hinges on the f a c t that every decreasing sequence of o r d i n a l numbers i s e v e n t u a l l y constant. Let any cover be indexed by w e l l ordered set S . Prom the cover [C_,"L _ c s e l e c t a subset B by the r u l e S S £ o B = {C : C szf C. f o r a l l t < s i n S] . s s x> B i s a cover f o r X , and every sequence of decreasing sub-sets of B w i l l correspond to a decreasing sequence of o r d i n a l numbers. So B possesses the descending chain c o n d i t i o n . For each p o s i t i v e i n t e g e r n the open sets of diameter l / n c o n s t i t u t e a cover f o r the space, hence we may apply the lemma to o b t a i n K n , a cover of the space which s a t i s f i e s the descending chain c o n d i t i o n and c o n s i s t s of open sets of diameter l / n . Denote by K the c o l l e c t i o n of a l l of the elements of a l l of these covers . Since K i s a c o l l e c t i o n of open covers of X by sets of diameter l / n , K i s a base for X . We complete the f i r s t h a l f of the proof by showing c o n t r a d i c t i o n upon the assumption that X i s not subcompact with respect to K . To t h i s end assume that there i s a r e g u l a r f i l t e r base i n K which i s not preconvergent. I t i s c l e a r that each member of t h i s r e g u l a r f i l t e r base must p r o p e r l y c o n t a i n some other member; i n p a r t i c u l a r , l e t 1 j_ be a p r o p e r l y decreasing sequence of f i l t e r base elements. Because each - 9 -of the f a m i l i e s s a t i s f i e s the descending chain c o n d i t i o n , only f i n i t e l y many of the sets "v\ can be s e l e c t e d from K n f o r each n = 1,2,... . Thus (v"^}^ ™ ^ n o t o n l y a decreasing sequence of subsets In complete space X but the diameters of the sets must tend to zero. We conclude that the i n t e r s e c t i o n of the c l o s u r e s of the ' i s nonvoid. Let x be some p o i n t i n t h i s i n t e r s e c t i o n and suppose that U i s a member of the given r e g u l a r f i l t e r base such that x' jL U . Since the f i l t e r base i s r e g u l a r we may s e l e c t a U' i n the f i l t e r base w i t h U' c U . C l e a r l y x £ U' , so by the s e p a r a t i o n i n the space X we may f i n d an i n t e g e r j s u f f i c i e n t l y l a r g e such that V . n U' = 0 . This gives the J c o n t r a d i c t i o n i n our f i l t e r base: V . fl U 1 = 0 . From t h i s 3 we conclude that x ' . i s an element of every member of the given r e g u l a r f i l t e r base; hence the f i l t e r base i s preconvergent and X i s subcompact w i t h respect to K . Before proceeding w i t h the d e t a i l s , we w i l l o u t l i n e our procedure f o r showing a subcompact m e t r i z a b l e space to be complete. Given m e t r i z a b l e space X , the metric completion of X i s a metric space X such that X can be mapped onto a dense subset of X by a metric p r e s e r v i n g homeomorphism. Considering X as a subspace of X , f o r each subset U open i n X there i s a subset tt open i n X such that U = tj n X , and the c l o s u r e of U i n 1 contains tt . I t i s w e l l known that a Gg-set i n a complete metric space i s a complete metric subspace [2, p. 189 ] . In p a r t i c u l a r given a 'subcompact metric - 1 0 -space X , we s h a l l demonstrate that X can be w r i t t e n as a Gj-set i n X . Suppose that the metric space X i s subcompact r e l a t i v e to some open base A = {U 1 „ . In the metric S S G o completion X of X we can f i n d a U open i n X such that U_ = U H X , f o r each s € S . Let t h i s he done f o r s s each s e S and define A = {U } „ . Each U has a S S € o S non-negative diameter, and since U i s dense i n U , s s diameter of U c equals diameter of U f o r each s i n S . s s For each p o s i t i v e i n t e g e r i l e t 0 ^ be the union of a l l the sets^ u\ i n % whose diameters are l e s s than 1 / i . Then the sets CL are open i n X and n 0 . 'is a G. set i n X . 1 1 = 1 1 6 For each i , the space X i s covered by base elements of diameter l e s s than l / i . Since U g cz ff f o r a l l s we have X c 0' f o r each p o s i t i v e i n t e g e r iv.., and i n p a r t i c u l a r Xa 0 i ' 1 = 1 To show that X = 0j_ we w i l l show the reverse i n c l u s i o n . , i = l ° 5 „ „ ~ Take x e Pi 0 . and l e t Vn be any element of A 1 = 1 1 1 of diameter l e s s than one w i t h x e . Such a e x i s t s because x e 0 ^ . Let n be an i n t e g e r greater than one and assume that the have a l l been s e l e c t e d f o r i l e s s than n , subject to the c o n d i t i o n s (a) x e V. , (b) "v\ e A w i t h diameter of l e s s than l / i , (c) V i c ( c l o s u r e taken i n X). Now x i s an i n t e r i o r p o i n t of V n so the distance from x n-1 to the complement of v"n ^ i s some nonzero number,L d . Let -11-V be any element of A which contains x and has diameter l e s s than minimum { l / n , d/J>) • Such a V e x i s t s because CO „ x e fl 0. assures us that we can f i n d elements of A 1=1 1 c o n t a i n i n g x and having diameter l e s s than 1 / i f o r "each p o s i t i v e i . The c o n d i t i o n that the diameter of V i s l e s s n than d/3 assures that x € v n c V n 1 ' I n d u c t i v e l y we have constructed a sequence of elements of A s a t i s f y i n g ( a ) , ( b ) , and (c) above f o r a l l i greater than 1 . Note f i r s t that the sequence ^ / • l ^ _ i s a decreasing sequence of nonvoid subsets of X w i t h diameters tending to zero. Since X i s complete Hausdorff space, the i n t e r s e c t i o n of the V\ must be e x a c t l y one p o i n t . A f o r t i o r i , {x} = n V . i = l 1 R e c a l l now t h a t . f o r every element of A , and i n p a r t i c u l a r f o r the sets V. we have V. 0 X = V. •% where V. i s an element of the given b a s i s A . C o n d i t i o n (c) can be r e l a t i v i z e d to the space X : Y7 a V . n f o r each i g r e a t e r than one. 1 x-1 CO CO Thus ( V -}. , i s a r e g u l a r f i l t e r base i n A and fl V. ^ 0 . co — 1 = 1 • 1 CO • But t h i s i n t e r s e c t i o n i s contained i n fl V. because i = l 1 CO V. <= V. f o r each i . We conclude immediately: {x} = n V. ; ' i = l 1 and. t h e r e f o r e "x e X . So fl 0. c X . This concludes the i = l 1 proof that X = f*l 0. , a G.-set i n complete metric space X i = l 1 6 - 1 2 -Thus X i t s e l f i s m e t r i z a b l e In a complete manner. In the above theorem the proof that every subcompact metric space i s complete only required the assumption that every countable r e g u l a r f i l t e r base i n A was preconvergent. Since a subcompact space i s t r i v i a l l y countably subcompact, the f o l l o w i n g theorem i s seen to be true. THEOREM ( 2 . 6 ) : In a me t r i z a b l e space the f o l l o w i n g are equivalent: ( i ) subcompactness, ( i i ) countable subcompactness, ( i i i ) completeness i n a s u i t a b l y chosen m e t r i c . The metric s e l e c t e d according to pa r t ( i i i ) w i l l of course give r i s e to an equivalent topology i n the space. De Groot sta t e s a form of the B a i r e category theorem i n a s e t t i n g where the c o u n t a b i l i t y c o n d i t i o n s are replaced by co n d i t i o n s w i t h an a r b i t r a r y i n f i n i t e c a r d i n a l , ]A as a parameter. S p e c i f i c a l l y : D e f i n i t i o n . Let be an i n f i n i t e c a r d i n a l . ' A (e.g. closed) subset S of a t o p o l o g i c a l space T i s c a l l e d ]A-thin, i f the i n t e r s e c t i o n of any f a m i l y of l e s s than yi op.en subsets of T i s not ( f u l l y ) contained i n S , unless t h i s i n t e r s e c t i o n i s empty. Complementarily, a (open) set 0 i n T i s c a l l e d jyT-puffed, i f the i n t e r s e c t i o n of any f a m i l y of l e s s than. ]A open subsets of T meets 0, unless t h i s i n t e r s e c t i o n i s empty. So the complement of an'K-puffed set i s J^-thin and conversely. For the countable case we have THEOREM ( 2 . 7 ) : X A subset i s T^-thin i f f i t i s a boundary set. ' Proof: Let U be a n l ^ o - t h i n subset o f t o p o l o g i c a l space X. We must show that U has v o i d i n t e r i o r . But the i n t e r s e c t i o n of l e s s than {Vi0 open sets i s a f i n i t e i n t e r s e c t i o n of open s e t s , -13-which i n t e r s e c t i o n i s an open set. So U contains no open subset and hence has v o i d i n t e r i o r . Conversely i f U i s a boundary set i n X then i t contains no nonvoid open set; hence i t does not co n t a i n a f i n i t e i n t e r s e c t i o n of open sets i m l e s s that i n t e r s e c t i o n i s v o i d . Hence U i s - t h i n . COROLLARY (2.8): ( i ) A closed subset i s Wo - t h i n i f f i t i s nowhere dense. ( i i ) An open set i s f\!0 -puffed i f f i t Is everywhere dense. Proof:" We' r e c a l l that a set i s nowhere dense i f f i t s c l o s u r e i s a boundary set. A l s o , the complement of an everywhere' dense set i s nowhere dense and conversely. A t o p o l o g i c a l space i s c a l l e d an yi-Baire space i f i t i s not the union of at most yi closed j4-thin subsets. With t h i s d e f i n i t i o n an 7\j0 - B a i r e space corresponds w i t h B a i r e space as defined p r e v i o u s l y . De Groot s t a t e s both an yi-Baire theorem and its s p e c i a l i z a t i o n to the countable case. We w i l l not repeat h i s proof of the >1-Baire theorem but w i l l supply the proof of the countable case which he omits. THEOREM (2.9): (Baire-de Groot): A subcompact r e g u l a r space i s an yi-Baire- space f o r every i n f i n i t e c a r d i n a l yi . THEOREM (2.10): ( B a i r e - deGroot): A countably subcompact r e g u l a r space i s a B a i r e space. Proof: Let there be given a t o p o l o g i c a l space X which i s subcompact w i t h respect to base A , and l e t { U ^ } b e a sequence of nowhere dense subsets of X . I f 0 i s any •14-open subset of the space we must show that 0 jzf U U. . i - 1 1 Since i s nowhere dense there e x i s t s some p o i n t x / 0\b"^  . Since X i s r e g u l a r we can f u r t h e r f i n d some open neighborhood of x whose c l o s u r e does not meet U^. Now A i s a base f o r the space, so s e l e c t some non v o i d base element V-^  e A such that x e V"1 c 0 and n = 0 . Take some i n t e g e r • n greater than one and suppose that f o r each j = 1 , 2 , . . . n - 1 we have determined base elements V- e A subject to the c o n d i t i o n s • J (a) V. c V . ^ , . ( b ) n v / 0 , i = i x (c) v. n u . = 0 . We would s e l e c t a V s a t i s f y i n g ( a ) , ( b ) , and (c) . n-1 n Now fl V. i s an open set which by v i r t u e of the i = l 1 i n d u c t i o n hypothesis i s nonvoid. U i s a nowhere dense set n - 1 _ 5 so there i s a p o i n t x e ( Pi V. )\u . Since X ^ i s r e g u l a r i = l 1 ' n we can a l s o f i n d some open neighborhood.of x whose c l o s u r e does not meet IT" . S e l e c t a base element V e A w i t h _ _ n _ n-1 n V n U = 0 and x e V c V c n V. . This V s a t i s f i e s xi xi 1*1 1*1 i 1 c o n d i t i o n s ( a ) , ( b ) , and ( c ) , a f o r t i o r i . CO The sequence a countable r e g u l a r f i l t e r base i n A ; since c 0 "and X i s subcompact w i t h respect to A we have 0 3 n V, / 0 1=1 1 -15-But from c o n d i t i o n (c) v/e see that ( n V. ) 0 ( U U.) = 0 . i = l "L j = l J We have thus produced a nonvoid subset of the a r b i t r a r y open set 0 which i s d i s j o i n t from the union of the closur e s CO of the nowhere dense sets U. . Hence U U. i s a boundary 1 i = l 1 set and the space X i s a B a i r e space.. (3) TOWARD A COMPARISON The c l a s s i c r e s u l t s of E. Cech achieve a B a i r e category theorem by means of a d e f i n i t i o n of completeness which i s a b i t more general than completeness as the term i s c u r r e n t l y used. The work of J . de Groot has accomplished a s i m i l a r end wit h a g e n e r a l i z a t i o n of compactness. The task of comparing these two approaches to the B a i r e category theorem i s , , u n f o r t u n a t e l y , only p a r t i a l l y accomplished i n t h i s t h e s i s . Subcompactness i s an h e r e d i t a r y p r o p e r t y under the formation of a r b i t r a r y C a r t e s i a n products (Theorem (2.1), i ) . Completeness i n the sense of Cech i s i n general only preserved^ by the formation of countable C a r t e s i a n products (Theorem (.1.3), i v ) . These general c o n s i d e r a t i o n s give b i r t h to the f o l l o w i n g example which demonstrates a r e g u l a r t o p o l o g i c a l space- which i s subcompact but not complete i n the sense of Cech. EXAMPLE (3.1): Let R be the r e a l numbers and f o r each N r e R l e t X be the h a l f open i n t e r v a l (0,1] w i t h the usual topology. Let X = P X w i t h the (usual) Tychonoff reR r topology. J c • -16-Each space X r Is a l o c a l l y compact Hausdor.ff space and, as a r e s u l t , subcompact (Theorem ( 2 . 3 ) ) . As was r e c a l l e d immediately above , X , being the C a r t e s i a n product of subcompact spaces* i s subcompact. To show that X i s not complete i n the sense of Cech we w i l l show that X cannot be a G -set i n some 6 c o m p a c t i f i c a t i o n . For each r e R l e t X^ be the closed i n t e r v a l [0,1] w i t h the usual topology. Then X = P X reR c i s a c o m p a c t i f i c a t i o n of X .. For an open set i n the Tychonoff product topology i t i s the case that the component p r o j e c t i o n s map onto the components w i t h at most a f i n i t e number of exceptions. So f o r a G^-set i t w i l l be the case that the component p r o j e c t i o n s w i l l map onto components, w i t h at most countably many exceptions. In p a r t i c u l a r i f X could be represented as a Gg-set i n X i t would be necessary that the component p r o j e c t i o n s of X i n t o the spaces X r would map onto a l l of the X w i t h at most countably many exceptions. This i s not the case and so X i s not/ a G f t-set i n X and, by d e f i n i t i o n , X i s not complete i n the sense of Cech. This example e s t a b l i s h e s that some subcompact spaces .are not complete i n the sense of Cech. To the converse question we can only answer that i n c e r t a i n s p e c i a l cases spaces complete i n the .sense of Cech are subcompact. In the case of m e t r i z a b l e t o p o l o g i c a l spaces completeness i n the sense of Cech and subcompactness are mutually equivalent to t o p o l o g i c a l completeness, hence equivalent to -17-each other. Another p a r t i a l r e s u l t i s THEOREM (3.2): Let X be a r e g u l a r t o p o l o g i c a l space. Then X i s subcompact i f there i s an open cover A^ of X such that f o r every f a m i l y of closed subsets {E_}_ „ w i t h s s e o (a) the f i n i t e i n t e r s e c t i o n property, (b) some element of {E } i s contained i n some element s of A1 , we have 0 E o / 0 seS 0 According to the Theorem (1.2) X i s complete i n the sense of Cech i f f i t has T-^i. sepa r a t i o n and has a countable f a m i l y of open coverings such that f a m i l i e s of closed subsets w i l l have nonvoid i n t e r s e c t i o n i f they s a t i s f y c o n d i t i o n s (a) and (b) f o r a l l i = 1,2,... . Proof: Let X be a r e g u l a r t o p o l o g i c a l space and A^ an open cover of X as described i n the hypotheses of the theorem. For each p o i n t x e X suppose ¥ i s an open set i n A, c o n t a i n i n g x . Define\ A tojbe the f a m i l y of a l l subsets of X which can be w r i t t e n i n the form W H O where x e X and 0 i s open i n X . We s h a l l prove that X i s subcompact w i t h respect to t h i s open b a s i s , A . Suppose that F = {U } 0 i s a r e g u l a r f i l t e r S S € O base i n A . For each U e F s e l e c t some U 1 e F w i t h s s U • c UJ c U . Define V = LP . S S S S S f" [V }- o i s a f a m i l y of cl o s e d subsets of X . S S € a To see that i t has the f i n i t e i n t e r s e c t i o n p r o p e r t y , we note that i f H V = 0 , then since each V i s the c l o s u r e of i = l ' s i -18-an element of F (namely U' ) we have s i n n n n u' c n u' = rv v r = 0 , S S cj ' J 1=1 " i 1=1 i i = l 1 a c o n t r a d i c t i o n . By the way_the.._base A v/as constr u c t e d , c o n d i t i o n (b) i s s a t i s f i e d a f o r t i o r i . By the hypotheses of the theorem concerning the open cover A^ v/e conclude that fl V A 0 . But then since V c U f o r each s e S , seS 5 3 n u r> n v { 0 seS s seS s showing X to be subcompact w i t h respect to the open b a s i s A J-l . COROLLARY (J>.J>): Let X be a r e g u l a r t o p o l o g i c a l space. Then X i s subcompact i f there i s a f i n i t e f a m i l y of open covers { A ^ } / ^ of X such that f o r every f a m i l y of closed subsets [E 3 c w i t h 1 s J s e S (a) the f i n i t e i n t e r s e c t i o n property, (b) f o r each i = 1,2,... '', n there i s some U. e (E ] and some open subset e A^ w i t h c , we have H E d 0 . seS S Proof: The extension to the case of f i n i t e l y many open coverings i s not e s s e n t i a l l y d i f f e r e n t from the main theorem. Set B = {V, n V 0 n . . . n V Iv. e A, f o r each 1 2 n 1 i l w 1 -19-Then B i s a s i n g l e covering of the space and everything reduces to the previous theorem w i t h B p l a y i n g the r o l e of A j • Indeed suppose v/e have a f i n i t e f a m i l y of open coverings of X. and a f a m i l y of closed subsets s a t i s f y i n g c o n d i t i o n s (a) and (b) . So we have l ^ ) ^ c 1ES1 a n d • w i e A i w i t h U. c ¥. f o r each i = 1,2,... , n . Having assumed the f i n i t e i n t e r s e c t i o n property we see that h n 0 £ n U. c fl W. e B . i = l 1 i = l 1 S u b s t i t u t i n g f o r {E } the f a m i l y of a l l f i n i t e i n t e r s e c t i o n s s of elements i n [ E l , theorem (3.2) a p p l i e s . . We s t a t e e x p l i c i t e l y the a p p l i c a t i o n of these r e s u l t s to spaces complete i n the sense of Cech. COROLLARY (3-^): I f X^ i s a t o p o l o g i c a l space complete i n the sense of Cech and_the_equivalent c o n d i t i o n s t a t e d i n theorem (1.2) can be achieved wl.th only a f i n i t e f a m i l y of open coverings, then X i s subcompact. Cases to which c o r o l l a r y (3.4) apply e x i s t . EXAMPLE (3-5)' Let Q . denote the r a t i o n a l numbers and l e t X = (0,1 )\Q w i t h the topology of the E u c l i d i a n m e t r i c . For each r a t i o n a l number q define V = (0,l)\{q} . X = fl V q qeQ. q i s a r e p r e s e n t a t i o n of the metric space X as a G^-set i n the complete metric space [0,1]. Then i t i s w e l l known that *- / -20-t h i s i s s u f f i c i e n t f o r X to be t o p o l o g i c a l l y complete [2, p. 189]; so X i s both subcompact and complete i n the sense of Cech. We a s s e r t that c o r o l l a r y (3.4) does not apply to X . To demonstrate t h i s we w i l l show that every open subset of X contains a decreasing sequence of closed sub-'sets w i t h v o i d inters-ec-ti-on. To t h i s end v/e consider X as a subspace of the u n i t i n t e r v a l (0,1) w i t h the induced top-ology. Any open set i n X contains the i n t e r s e c t i o n of X v/ith some nondegenerate i n t e r v a l . I f q i s a r a t i o n a l p o i n t i n t e r i o r i n t h i s i n t e r v a l (speaking of q has meaning i n the space (0,1)), then e v e n t u a l l y the subsets defined to be X H [q, q + l / i ] w i l l be contained i n t h i s i n t e r v a l ; i f N i s the f i r s t i n t e g e r f o r which t h i s i s t r u e , then [v\ 0 w i l l be a decreasing sequence of subsets c l o s e d i n X and having v o i d i n t e r s e c t i o n . (k) RANKED SPACES OP KUNUGI The Japanese mathematician K. Kunugi has proved a form of the B a i r e category theorem which does not assume an axiom of s e p a r a t i o n . In a d d i t i o n the theorem of Kunugi allows i n some cases the c o n c l u s i o n that more than countably many open everywhere dense subsets have dense i n t e r s e c t i o n . This g e n e r a l i z a t i o n i s obtained without strengthening the d e f i n i t i o n of everywhere dense set as was the case v/ith the j4-puffed sets defined by J. de Groot. The work exposited here i s to be found i n [5 ] . Before p r e s e n t i n g the d e f i n i t i o n s of the s t r u c t u r e s -21-used by Kunugl we r e c a l l that P. Hausdorff formulated these three neighborhood axioms i n h i s c l a s s i c t e x t [4, p. 2591' (a) Every p o i n t x has at l e a s t one neighborhood U ; X and U v always contains x . (b) For any two neighborhoods U and V of the same p o i n t , there e x i s t s a t h i r d , W a U n V ^ ' ' X X X (c) Every p o i n t y e U has a neighborhood U c u x y x Pro f e s s o r Kunugi deals w i t h a p o i n t set X together w i t h a system of (open) neighborhoods s a t i s f y i n g axioms (A) and (C) of Hausdorff. A decreasing sequence ( p o s s i b l y t r a n s -f i n i t e ) of neighborhoods of a p o i n t x e X i s maximal i f the i n t e r s e c t i o n of a l l neighborhoods i n the sequence does not co n t a i n a neighborhood .of x . Denoting a sequence of neighborhoods by {V (x)} 0 < s < B , the o r d i n a l number S i s the type of the sequence. We define the depth of the space X at the p o i n t x to be the l e a s t o r d i n a l number W(X,x) f o r which x has a maximal sequence, subject to the two conventions: ( i ) i f axiom (B) of Hausdorff i s not s a t i s f i e d at some p o i n t x , W(X,x ) i s equal to zero; ( i i ) i f some p o i n t has no maximal sequence (i.e.- the p o i n t has a smallest neighborhood) then the depth at that p o i n t Is equal to the f i r s t o r d i n a l of potenc 2X . W(X) = i n f (W(X,x) : x e X] i s the rank of the space. The space X i s ranked i f f o r some l i m i t o r d i n a l number b <_ w(W)" , there e x i s t s a f a m i l y of open coverings B = {A }a < b indexed by the o r d i n a l s l e s s than b such that: -22-(a) Let there be given x an a r b i t r a r y p o i n t of X and V(x) i t s neighborhood, and an o r d i n a l l e s s than b . Then there e x i s t s some o r d i n a l a' wi t h a < a' < b and some subset U e A ' e B with U c V(x) . The o r d i n a l number b of t h i s d e f i n i t i o n i s c a l l e d the i n d i c a t o r of ranked space X . Note that i f the space X i s not a t o p o l o g i c a l space ( i . e . axiom (B) f a i l s at some p o i n t ) then b <_ w(X) = 0 ; i f X i s a t o p o l o g i c a l space then w(X) _> , the o r d i n a l of the n a t u r a l numbers. An open OwiC'jy? set has rank a i f i t i s a member of the cover A e B , a and i f t h i s i s not the case f o r any A ' , a' > a . a Let V a ( x & ) (a < c) be a decreasing sequence of neighborhoods where c < b , the i n d i c a t o r of X . The sequence i s fundamental i f the rank of each of the neighborhoods V (x ) i s i n c r e a s i n g and f o r each a < c there i s an a' , a x a 3 a < a' < c such that x . = x ' , and the rank of V ,(x ,) — a a -|- J - a a i s s t r i c t l y l e s s than the rank of V , -, (x , , ) . A ranked .v a ' + l a ' + l space i s complete i f every fundamental sequence has nonvoid i n t e r s e c t i o n . THEOREM (4.1): Every complete metric space can be given the s t r u c t u r e of a complete ranked space. Proof: This theorem can be approached v i a the theory of uniform spaces. Take some metric d f o r which the space X i s complete and co n s t r u c t a uniform s t r u c t u r e f o r the space i n the usu a l manner. A base f o r t h i s u n i f o r m i t y i s given by the f a m i l y C(x,y) e X x X : d(x,y) < i}£ . - 25 -X thus takes on the s t r u c t u r e of a complete uniform space [2, p. 33o], which can a l s o be looked upon as a rank s t r u c t u r e w i t h i n d i c a t o r & . In a complete uniform space every Cauchy f i l t e r i s convergent [2, p. 341] . R e c a l l that a f i l t e r i n X i s ' Cauchy i f f o r each entourage of the diagonal some member of the f i l t e r i s contained i n a neighborhood generated by the entourage of the diagonal. Let £ Vi( xi^i-l ^e a fundamental sequence of neighborhoods-of the ranked space. The c o n d i t i o n s that the ranks of the neighborhoods V j _ ( x j ) must e v e n t u a l l y grow a r b i t r a r i l y l a r g e imply that f o r each p o s i t i v e i n t e g e r n some neighborhood v _ _ ( n ) ( X j _ ( n ) ^ ^ s contained i n an open b a l l of diameter i / n : V i ( n ) ( X i ( n ) ; ) c <y : d ( x i ( n ) ^ ) < 1 / n } ' Let F denote the (Cauchy) f i l t e r generated by {V\(x.l} . Then by the completeness of X as a uniform space we see CO that 0 V.(x.) 3 n W / 0 . Hence the fundamental sequence i = l 1 1 WeF ./ { v \ ( x ^ ) l has nonvoid i n t e r s e c t i o n and so X i s complete as a ranked space. THEOREM (4.2): (Baire-Kunugi): Let X be a complete ranked space w i t h i n d i c a t o r o r d i n a l b . Then the i n t e r s e c t i o n of any nonvoid f a m i l y of open everywhere dense subsets, indexed by the o r d i n a l s l e s s than the o r d i n a l d <_ b , i s everywhere dense. Proof: Given a sequence of open everywhere dense subsets -25-{Ba3 (o <_ a < d) we can, f o r t e c h n i c a l reasons, compress i t by d e f i n i n g f o r each even a < d , A = B D B n . a a a+1 The sets A w i l l each be open everywhere dense subsets, a Suppose U i s some nonvoid open subset of X . To •show that U fl ( 0 B ) / 0 i t w i l l now s u f f i c e to show that a<d a U n {n A a : a< d , a even] jL 0 . We begin the ( t r a n s f i n i t e ) i n d u c t i o n by s e l e c t i n g a a some x Q e U . By the d e f i n i t i o n of ranked space there i s a rank c and an open neighborhood V (x ) of rank c o o v o o w i t h V (x ) c (U 0 A ) . Let a < d be given and suppose o o o , that f o r each e < a there has been defined a p o i n t x , r e a rank c , and a neighborhood V (x ) of rank c wi t h e 3 e x e e the neighborhoods decreasing and the ranks i n c r e a s i n g . Suppose f u r t h e r that f o r each even o r d i n a l e < a , x = x , , e e+1 c < c , , and V (x ) c A e e+1 3 e v _er e Suppose f i r s t that a i s an even n o n - l i m i t ordinaL. Set TJ = V , (x , ) and l e t x be any p o i n t i n the open a a—± a—x a nonvoid set A fl U . This set i s nonvoid because A i s cX 3, cl everywhere dense. I n the case that a i s a l i m i t o r d i n a l set U = fl V_(x Q) . U i s nonvoid i n v i r t u e of the a . Q e e a e< a • f a c t that { V e ( x e ) } (e < a) i s a fundamental sequence i n the complete ranked space X . We a s s e r t f u r t h e r that U a i s open. Indeed i f y e D V (x ) f o r each e we can e<a e e' . f i n d a neighborhood V ,(y) w i t h ranks i n c r e a s i n g , the -26-neighborhoods V ,(y) decreasing, and V ,(y) c V (x ) . Then {V e,(y)3 (©' < a) w i l l be a decreasing sequence of neighbor-hoods of type les's than than the depth-of the space w(X) ; i n p a r t i c u l a r t h i s sequence cannot be maximal. We conclude that there i s some neighborhood V'(y) w i t h y e V»(y) c n V (y) c U . e<a e a So U i s open and we may ther e f o r e designate by x some a • a p o i n t i n the nonvoid open set A fl U a a In both cases the c o n s t r u c t i o n performed immediately above can be accomplished f o r the case y = x to f i n d a a neighborhood V (x_,) of rank c w i t h c > c and ^ a a a a e V" (x ) cr V" (x ) f o r a l l e < a . In p a r t i c u l a r , V c A . a a e v e r a a . F i n a l l y i f a i s some odd o r d i n a l set x = x -, a a-1 and s e l e c t according to the d e f i n i t i o n of ranked space a rank c > c n and a neighborhood V ( x 0 ) o f rank c w i t h a a—x Y a a a • V (x ) c V , (x n ) . aK a- a - l v a - l y that Thus we have a fundamental sequence {V (x )} such a a a<d V a ( x a ) c U n ^ n A a : a < d , a even] . a Since X i s complete t h i s fundamental sequence has nonvoid intersecti-on-an-d--so' D B fl U i s nonvoid, the a<d a " des i r e d r e s u l t . c I t w i l l be r e c a l l e d that ranked spaces need not be t o p o l o g i c a l spaces. But i n the case t h a t axiom (B) - 2 7 -of Hausdorff f a i l s f o r any p o i n t of the space the above theorem reduces to the a s s e r t i o n : In a complete ranked space every open everywhere dense set i s everwhere dense. This becaaue the i n d i c a t o r of a n o n - t o p o l o g i c a l ranked space i s defined I D be zero. COROLLARY (4.3): A ranked t o p o l o g i c a l space i s a B a i r e space. Proof: A f o r t i o r i . We c l o s e t h i s s e c t i o n w i t h the f o l l o w i n g example demonstrating a t o p o l o g i c a l space which i s a complete ranked space but i s n e i t h e r complete i n the sense of Cech nor subcompact. EXAMPLE (4.4): Let X c o n s i s t of the r e a l numbers w i t h the c o f i n i t e topology. That i s , every open set i s a subset of X which i s the complement of some f i n i t e p o i n t set i n X . X i s a T^ separated space but i s not . T^ ; s o / i s n e i t h e r complet i n the sense o£ Cech nor subcompact. Define U ( x1, x2, . . . , x n , x,n) = xACx^, . .., x n ] ; where n i s a p o s i t i v e i n t e g e r , x e X , and x^ i s a p o i n t of X d i f f e r e n t from x f o r each i = 1 , 2 , . . . , n . Further / d e f i n i n g A n = {U(x 1, x n , x,n) : x n e X ; n f i x e d ] , X i s seen to be a ranked space w i t h ranking s t r u c t u r e B and i n d i c a t o r <jj . Indeed B contains every open set and given - 2 8 -U(x-^, . .., x n , x, n) an open neighborhood of x and m > 0 , set p = maximum {n + 1 , m] . S e t t i n g x = x , = . . . = x we have x e \](x^,;.., x^, . .., x^, x, p) c U ( x 1 , ..., x n , x, n) . For the completeness note that the i n t e r s e c t i o n of any countable sequence of subsets i n B w i l l have the c a r d i n a l i t y of the continuum; a f o r t i o r i such an i n t e r s e c t i o n i s nonvoid and X i s a complete ranked space. (5) CONCLUDING REMARKS A f o u r t h approach to the problem of u n i f y i n g the B a i r e category theorem i s given by E. E l i a s Zakon i n [ 8 ] . There i s presented a B a i r e - l i k e theorem i n the context of uniform t o p o l o g i c a l spaces [ 8 , theorem 5-1, p. 3 8 3 ] . L i k e the work of de Groot, t h i s paper attempts to formulate a theorem which allows f o r uncountable unions. While the work of Zakon i s i n t e r e s t i n g i n the cases of higher card-i n a l i t y , i t dbes not s p e c i a l i z e to the usual B a i r e theorem. A p p l i e d to the r e a l l i n e w i t h the usual topology, Zakon's theorem gives conclusions a p p l i c a b l e only to the coarser i n d i s c r e t e topology (the whole space and the n u l l set being the only open s e t s ) . Completeness i n the sense of Cech i s defined i n the s e t t i n g of Tychonoff spaces; but the proof of the B a i r e theorem only r e q u i r e s r e g u l a r i t y . If i n the s e t t i n g of r e g u l a r spaces one uses the equivalent i n t r i n s i c c o n d i t i o n of Theorem ( 1 . 2 ) as the d e f i n i t i o n , then one achieves a modified d e f i n i t i o n of completeness f o r which a l l of the r e s u l t s presented - 2 9 -i n t h i s paper remain true. Using the f a c t that a complete ranked space need not have any separation p r o p e r t i e s i t was not d i f f i c u l t to c o n s t r u c t a space which was a complete ranked space but n e i t h e r subcompact nor complete i n the sense of Cech. I t seems that a very p e r t i n e n t question i n t h i s regard would be the r e l a t i o n s h i p among spaces complete i n the sense of Cech, spaces subcompact, and r e g u l a r spaces w i t h a completely ranked s t r u c t u r e . Conspicuously l a c k i n g from s e c t i o n 4 i s the theorem which assures that every l o c a l l y compact Hausdorff space can be given the s t r u c t u r e of a complete ranked space. This theorem i s announced i n [ 5 ] and a proof i s sketched i n [ 6 ] ; the present w r i t e r has been unable to v e r i f y that proof. Y. Yoshida, a student of Kunugi, has proved t h i s theorem only a f t e r strengthening the hypotheses [7]. A l l of the paper*encountered i n the research f o r t h i s t h e s i s d e a l t w i t h the task of f i n d i n g s u f f i c i e n t c o n d i t i o n s that a space be a B a i r e space. A f i n a l f u l l y / / u n i f i e d B a i r e theorem would be achieved by d e f i n i n g some property-property B-which would allow the theorem: Let X be a r e g u l a r t o p o l o g i c a l space; then X i s a B a i r e space i f f X has property B . The r e s u l t s of Kunugi give hope that the assumption of r e g u l a r s e p a r a t i o n could be e l i m i n a t e d . X . -30-REFERENCES [1] Cech, On bicompact spaces, Annals of Mathematics 3 8 (1937), 8 2 3 - 8 4 4 . - — [ 2 ] Engelking, R., Out l i n e of general topology, North-Holland, Amsterdam ( 1 9 6 8 ).. [3] deGroot, J . , Subcompactness and the Ba i r e category theorem, K o n i n k l i j k e Nederlands Akademie Van Wetenschappen Proceedings, Series A, Mathematical Sciences 66 (1963), 76I-767. [4] Hausdorff, F., (J.R. Anman, et a l . , Tr.) Set theory, Chelsea, New York (1957). -[ 5] Kunugi, K., Sur l e s espaces complets et regulierement complets I , Proceedings of the Japan Academy 30 (1954), 5 5 3 : 5 5 b . [6] _, Sur l e s espaces completes et regulierement . complets I I , I b i d . , 912-916. [7] Yoshida, Y., Compactness and completeness• i n . ranked spaces,- '. Proceedings of the Japan Academy 4 4 ( 1 9 6 8 ) , 25 1-25 4 . , - . [ 8 ] Zakon, E., On uniform spaces w i t h quasi-nested baW, Transactions of the American Mathematical. S o c i e t y 1 3 3 -(1968), 3 7 3 - 3 8 5 -

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