REMOTE POINTS IN PR AND P-POINTS IN 0R - R by CHI-MING LEUNG B.Sc, New A s i a College, The Chinese University of Hong Kong, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF J MASTER OF^ARTS i n the Department Qf MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1971 In presenting an advanced degree at the U n i v e r s i t y the Library this shall I f u r t h e r agree for thesis in partial make i t f r e e l y that permission h i s representatives. of this written thesis MfcRe^cpt-f C The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a D a t e gain ApRiL S Columbia i 4 • 14 7 1 I agree that copying o f this thesis by t h e Head o f my D e p a r t m e n t o r I t i s understood f o r financial Columbia, f o r r e f e r e n c e and s t u d y . f o rextensive permission. Department o f of B r i t i s h available s c h o l a r l y p u r p o s e s may be g r a n t e d by f u l f i l m e n t o f the requirements f o r shall that copying o r p u b l i c a t i o n n o t be a l l o w e d w i t h o u t my Supervisor : Professor J.V. Whittaker ABSTRACT We are going to study the remote points i n in PR — R . A remote point i n chosure of any d i s c r e t e subset of of $R—R i f every gR 0R and the P-points i s a point which i s not i n the R . G^-set containing A point p p e *R - R gR i s a P-point i s a neighbourhood of p . I ) ACKNOWLEDGEMENT S • ' I am deeply indebted to Professor J.V. Whittaker for suggesting the topic of this thesis and for rendering invaluable assistance and encouragement throughout the course of my work. I would like to thank Professor T.E. Cramer for reading the f i n a l form of this work. ! 1 TABLE OF CONTENTS INTRODUCTION CHAPTER I : Preliminaries 1 CHAPTER II : g-subalgebras 8 CHAPTER III : The A-points of " X - X . 1 5 CHAPTER IV : Remote points i n 3R 21 CHAPTER V : Prime ideal structure and remote points 28 BIBLIOGRAPHY 37 As we know, every completely regular space X has a compactification 3X such that every function f^ i n C(BX) . This thesis i s devoted to study the papers f i n C*(X) has an extension to a function [1], [3], [4], [5]. In chapter II, we study the class of subalgebras of B-subalgebras. in With each 3-subalgebra A of C(X), C(X) called we define A-po'ints 3X — X . Then we study the A-points i n chapter III . In chapter IV, we turn our attention to the remote points i n prime ideal structure of C(X) . 6R . Finally, we study the CHAPTER I PRELIMINARIES Throughout this thesis, a l l given spaces are assumed to be completely regular and Hausdorff. C(X) w i l l denote the collection of a l l real-valued continuous functions on X, and C*(X) w i l l denote the subcollection of bounded functions. Under the pointwise operation, commutative rings with identity. C(X) and C*(X) are A l l ideals i n C(X) or C*(X), w i l l always mean proper ideals. If S i s a set, then unmodified, | s | w i l l denote the cardinality of S. As i s standard, l e t c denote the cardinality the continuum. If Furthermore, we assume the continuum hypothesis S C " » then C1 S, X int S, 8S x (c = V^i ). w i l l denote, respectively, the closure, X interior and boundary of S i n X. 2^° of If f i s a function, then we l e t f < — denote the inverse map. Definition (1.1) For f e C(X), is called a zero set i n X while The family Z[C(X)] Z(f) = f X — Z(f) < _ (0) = { x e X : f(x) = 0 } i s called a cozero set i n X . of a l l zero sets i n X w i l l be denoted by Z(X) . Remark (1.2) (1) The family Z(X) of a l l zero sets i s a base for the closed sets. (2) f i s a unit of C(X) i f and only i f Z(f) = <f> (3) Every zero set i s a Gg set . Definition (1.3) separated i n X f[A] = {0} , Two subsets A subspace in C*-embedded i n X C(S) of X are said to be completely f e. C*(X) S of X such that 0 <^ f <_ 1. i s said to be C-embedded i n X i f can be extended to a function i f every function i n C*(S) i n C(X) . S i s can be extend to a function C*(X) . Definition (1.5) A non-empty family F of Z(X) i s called a z - f i l t e r on provided that (a) <j> i F (b) i f Z ( f ) , Z(g) e F , then (c) i f Z(f) e F , Z(g) e If i n addition, F B f[B] = {1} . every function X and i f there exists a function Definition (1.4) in A F Z(X) Z(f) (\ Z(g) e F and Z(f)CZ(g), then Z(g) e F . i s not contained i n any other z - f i l t e r , then i s called a z-ultrafilter on . X . Theorem (1.6) (a) If I i s an ideal [resp. maximal ideal] i n C(X), then Z[I]={Z(f):fel} is a z-filter If F (b) [resp. z-ultrafilter] on is a z-filter is an ideal C [resp. z-ultrafilter] on X, then Z < - [F]={f:Z(f) e F} [resp. maximal ideal] i n C(X) . Hence the mapping in X . Z i s one-one from the set of a l l maximal ideals onto the set of a l l z-ultrafilters. Definition (1.7) implies An ideal I i n C(X) i s called a z-ideal i f Z(f) e Z[I] f e I. Definition (1.8) f A z-filter in X i s called a prime z - f i l t e r i f f has the following property : whenever the union of two zero sets belongs. to F i then at least one of them belongs to p . Definition (1.9) H Z[I] f <>j -. An ideal I in C(X) or Otherwise I C*(X) i s said to be fixed i f i s said to be free . Theorem (1.10) (a) The fixed maximal ideals i n C(X) Mp The ideals Mp = { f e C : f(p) = 0 } are distinct for distinct isomorphic with the real field p . For each onto The ideals Mp isomorphic with the real field C /M? is are precisely the sets p . For each R ; in fact, the mapping onto C/Mp Mp(f) —> f(p) = ' { f e C* : f(p) = 0 } are distinct for distinct the unique isomorphism of p, R . The fixed maximal ideals i n C*(X) Mj (p e X) . R ; in fact, the mapping i s the unique isomorphism of C/Mp (b) are precisely the sets R . (p e p, X) . C /M* i s M^(f) —> f(p) i s Definition (1.11) for which For p e X, l e t 0 p denote the set of a l l f i n C Z(f) i s a neighbourhood of p . If Mp = Op , then p is called a P-point of X . Remark (1.12) p p eX i s a P-point of X i f and only i f every containing i s a neighbourhood of p . Remark (1.13) (a) For p e X, Mp i s the only maximal ideal (fixed or free) containing Op • (b) If P i s a prime ideal i n C, and PCMp , then Definition (1.14) space i n which Theorem (1.15) X By a compactification of a space PDO p . X, we mean a compact i s dense. Every space X has a Stone-Cech compactification BX with the following equivalent properties : (1) (Stone) Every continuous mapping Y has a continuous extension T (2) (Stone-Cech) Every function function (3) from T from X 3X into Y. f i n C*(X) has an extension to a f& i n C(BX) . (Cech) Any two disjoint zero sets i n X have disjoint closures i n PX . (4) into any compact space For any two zero sets Zj and Z 2 in X , (5) If X i s dense and C*-embedded i n T, then X C T C£X . (6) If X i s dense and then gT = gX . Furthermore, gX C*-embedded i n T, i s unique, i n the following sense : i f a compactification T of X satisfies any one of the listed conditions, then there exists, a homeomorphism of gX onto T that leaves X pointwise fixed. Remark (1.16) (1) For S C X . S i s C*-embedded i n X i f and only i f C1 S = gS . DV PA (2) The mapping Theorem (1.17) f —> f^ i s an isomorphism of C*(X) onto C(gX) . The maximal ideals i n C*(X) are precisely the sets M* = P { f e C*(X) : f&(p) = 0 } and they are distinct for distinct (p e gX) , p . The maximal ideals i n C(X) are precisely the sets MP = { f e C(X) : p e C l ^ Z ^ f ) } and they are distinct for distinct Definition (1.18) M Let M R . If M pe gX) , p . be a maximal ideal of C(X). [resp. C*(X)] . i s said to be a real ideal i f C/M real f i e l d ( [resp. C*/M] i s not real, then we c a l l i s isomorphic to the M hyper-real . Definition (1.19) (a) X i s said to be realcompact i f every real maximal ideal i n C(X) i s fixed. (b) X By a realcompactification of X, we mean a realcompact space i n which i s dense. (c) X i s said to be pseudocompact i f C(X) = C*(X) . Theorem (1.20) i s hyper-real i f and only i f M** 5 i C Theorem (1.21) M p i s real. Let vX denote the set of a l l points p e BX vX i s the largest subspace of BX (b) vX i s the smallest realcompact space between particular, X Theorem (1.22) (1) Every (completely regular) space Every continuous mapping x from VX i s the Stone extension of T Every function C(vX) . X i s C-embedded. X and BX . In (Necessarily X has a realcompactification with the following equivalent properties. has a continuous extension x° from (2) i n which i s realcompact i f and only i f X = vX . contained i n BX, where T such that Then (a) vX, contains a unit of f in X into any realcompact space into into Y . (Necessarily, T ° = ^IvX, BY.) C(X) has an extension to a function f = f*|vX .) Furthermore, the space v Y vX f° i n i s unique, in the following sense : i f a realcompactification T of X satisfies any one of the listed conditions, then there exists a homeomorphism of vX T that leaves Theorem (1.23) of R, X pointwise fixed. If f e C(X) , and aR denotes the one-point compactification then there is a (unique) continuous function agrees with f Theorem (1.24) f* : BX —> aR which on X . In the ring C(X), and also i n C*(X), containing a given prime ideal form a chain. sets.) onto the prime ideals (A chain i s a totally ordered B-SUBALGEBRAS Let A be a commutative ring with an identity. Let F be the set of prime ideals i n A . For E c A, define V(E) = { P e F : ECp} Note that F (1) V(+) (2) V(A) = < f > (3) V( u E ) = H V(E ) ieJt ie£ = ± EC ± ± A where H (4) E cr A , V(E H F) = V(E) U V(F) Therefore the V's determine a topology on , ie £, i s an index set Fc A . F . This topology i s called the hull-kernel topology. Now for a e A , define V(a) = { P e F : aeP} and l e t F a = F — V(a) . Theorem (2.1) (1) { F topology. a : a e A } i s a basis of open sets for F with tae hull-kernel , (ii) F i s compact. (i) Let 8 be a closed subset i n F , then Proof : ECA . Now i f and only i f P 4 B Pe F- B and only i f there exists exists . a e E such that (ii) by E. We claim for some ae E Suppose Pe F such that a . Thus F = U F aeE I = A . Suppose P e F , then Pe F a a a i f and only i f there F— B = u aeE F a . Let I = (E) = ideal generated I ^ A , then by Zorn's lemma for some for some i f and only i f E <£ P i f P , ECA. B - V(E) a e E . . Hence IC P a 4 P • But a e E C I C P , contradicting a y P . Therefore we must have I = A . So r • . 1 = J ^ i i » a^ e E , b^ e A . Now for P e F , since 1 4 P > there a exists i , 1< i< n proves that such that a..- 4 P . It follows that 1 F = F a and F l U ••• U F d Pe F . This i . a 0 a n i s compact . Notation (2.2) Let denote the collection of maximal ideals i n A endowed with the hull-kernel topology. Definition (2.3) By a subalgebra - A of C(X), we mean a subalgebra i n the usual sense which contains the constant functions. Given a subalgebra A of C(X). Define for each p e BX , i II where = f* maps BX { f e A : (fg)*(p) = 0 for a l l g e A } into the one point compactification of R as stated in 1 . 2 3 . Let G Theorem (2.4) Proof : 0 e and 1 £ M^., 3 X } -. E p e 3X . we see that $ $ i s an ideal i n A . To prove that to show that i f f, g e A f, g e A , Then : p i s a prime ideal i n A, Since Obviously = ' { A choose with h, k e A f, g £ , such that (fghk)*(p) i 0 . Thus fg and ^ A . i s prime, i t suffices then fg 4 (fh)*(p) f 0 . Now l e t and (gk)*(p) ± 0 . MJ . ' A Definition (2.5) Let x : BX—> G^ be such that A said to be a B-subalgebra of C(X) if T a T (p)'= A i s a homeomorphism of M . A Remark (2.6) For C*(X) and C(X) are 3-subalgebras of C(X) . f e A , let S (f) A = T ~ { p e G a A : f = ' { p e PX : f e = n z((f )*) geA g . e P } } . A is gX onto Since i s closed i n 8X, Z((fg)*) is continuous, since S ( f ) is closed in 6X . Note that A { { p e : fe P } : fe A } i s a base for the closed sets i n G. A . Definition (2.7) if A subalgebra { Z(f*) : f e A } A of i s said to be B-determining C(X) forms a base for the closed sets in 6X . A i s said to be closed under bounded inversion i f f f e A with i s a unit of A f >_ 1 . Definition (2.8) An ideal I in A f e I whenever fe A and g e l i s said to be absolutely convex i f satisfying | f| P A M , M P , G , Theorem (2.9) T and [ g{ M. , M For convenience, we shall abbreviate to whenever , G A A , T. and A S , respectively. Given a subalgebra A of C(X) , the following are equivalent. i s B-determining (1) A (2) G i s Hausdorff and (3) T Proof : with x i s one-to-one i s a homeomorphism (1) implies (2) . Suppose A i s B-determining and l e t p,q e B p i q . By [2, 6.5(b)], there exists Zj U Z 2 = X and { Z(f*) : f e A } Z , Zj, e Z(X) 1 p 4 Clg^Z} , q gf C1^Z2 . Since A x such that i s B-determining, is a base for the closed sets in gX . So we can choose f, g e A p 4 Z(f*) Z> CI. „Z, such that and q ^ Z(g*) 3> CI. Z 9 choice above, -f ^ M M e G — { M e G : f e M } which i s an . Thus P P s O open set i n G . Similarly s s S M eG H — { M e G :ge M } which i s an open set i n G . Furthermore by the choice of f, g, we see that Thus and {M S G : f' e_ M 8 e } U { M e G :ge M s G—{M eG:geM } S s are arbitrary, G s s s are disjoint open sets i n G . Since i s Hausdorff. be a closed set i n BX . Since fg = 0 . } = G . So G — { M e G: f e M } Since M ±M P q , x (2) implies (3) . It suffices to prove that is continuous, . By the p A. pA BX i s compact, x[F] i s compact. Since G p, q i s one-to-one . x i s closed. Let F F i s compact . Since x is Hausdorff, x[F] i s closed. (3) implies (1) . Let F be a closed set i n BX and p e BX with p 4 F . Since x i s a homeomorphism, closed sets i n BX . Thus there exists F C S(f) . Since Thus { S(f) : f e A } i s a base for the fe A such that p 4 S(f) and S(f) = H Z((fg)*) , (fg)*(p) f 0 for some geA p V Z(f*) ; but F C S(f) C ge A. Z((fg)*) . This proves that {Z(f*): f e A} is a base for the closed sets i n BX . Theorem (2.10) Given a subalgebra A of C(X), the following are equivalent. (1) A i s closed under bounded inversion. (2) If I i s an ideal i n A, then (3) Every ideal i n A (4) , M C G A A . f\ Z(f*) f <p . fel i s contained i n some M . p jtaiiisM (5) Every Me i s absolutely convex (1) implies (2). Let I be an ideal i n A . Let F={ Z(f*):& 1} . Proof : To prove (2), by the compactness of 8 X , i t suffices to show that f has the f i n i t e intersection property. Let f , • • •, f e I and l e t g=ff + -•-+f$f I. n Then Z(g*) = (\ Z(ff) . Suppose Z(g*) = <j> . Then |g*(p)| > 0 for a l l i=l 1 p e 8 X . Since So n 8 X i s compact, there exists g >_ r, and g i s a unit of A . Since this i s a contradiction. So we must have r> 0 such that g e l and since |g*(p)| ±_ r > 0. I i s proper, Z(g*) = cj> . (2) implies (3) . Let I be an ideal i n A . Let p e H Z(f*). fel We claim that IC M . For i f f e l , then (fg)*(p) = 0, for a l l g e A . So p fe M P fg e I for a l l g e A . So . (3) implies (4) . Obvious. (4) implies (5). convex. Let f e A and for a l l h e A . Since h e A . So fe M p It suffices to show that ge M P X satisfying i s dense i n gX , M P | f | <_ |g| i s absolutely . Then | (fh)*| <_ | (gh)*| | fh| <^ | gh| for a l l . (5) implies (1). Since 1 does not belong to any maximal ideal, i t follows that f i s a unit of A This completes the proof . whenever f e A with f ^_ 1 . Theorem (2.11) Given a subalgebra A of C(X), the following are equivalent. (1) A (2) 'A Proof : i s a B-subalgebra of C(X) . i s B-determining and closed under bounded inversion. Suppose B-determining. A i s a B-subalgebra of C(X) . By 2.9, A i s By 2.10, A is closed under bounded inversion. Conversely suppose (2) holds. By 2.9, - BX G onto G . By 2.10, are comparable. C(X) . So M C G . Since G T is T M = G . This proves that i s a homeomorphism of 2 A , no two ideals of i s a B-subalgebra of THE A-POINTS OF Let A be a g-subalgebra of gX - X C(X). forms a base for the closed sets i n BX. feA, let S*(f) = S(f) (\ X* . Then Let By 2.9, the family X* denote { S*(f) : {S(f): feA} gX — X . For feA} i s a base for the closed sets i n X* . For convenience, let us agree that the symbols "CI" , " i n t " and " 9 " , without subscripts, refer to the topology of Definition (3.1) nonvoid Gg subset of Remark (3.2) X X i s said to have the Gg-property i f every has a nonvoid interior. Since i n a completely regular space a compact set has the A space S X* . X, contains a zero set containing S, every Gg containing i t follows that G^-property i f and only i f every nonempty zero set in X X has a nonempty interior. The following theorem w i l l be used several times throughout this thesis : Let Gg-property. Y be a nonvoid locally compact Hausdorff space with the If V i s a family of at most then H V then | n V\ >_ 2* i s dense i n Y. Definition (3.3) 1 . If, i n addition, Y Y, has no isolated points, ([5, 3.2]). Given a g-subalgebra said to be an A-point of V^x dense open subsets of X* i f , for a l l A of feA, C(X), a point p i 3S*(f) . p e X* is Remark (3.4) (1) A point p e X* neighbourhood of p (2) i s an A-point of X* whenever feA The set of A-points of X* Theorem (3.5) X is a such that Z e Z(BX) Proof : Suppose p e S*(f) . i s precisely f\ (X* — 3S*(f)) . feA i s realcompact i f and only i f for every X p e Z C M* P p e X* . Then contains a'unit C(X), i t follows that p e X* , there X* . is realcompact and by [2, 8.4] . By 1.20, unit of and i f and only i f S*(f) i s a Z(f^) C f of M P is hyperreal C(X). Since X* . By 1.17, f is a p e Z(f3) . This proves the necessity. Conversely, l e t p e X* . By assumption, there exists such that p e Z(g) C restriction of By 1.20, M P Theorem (3.6) then X* g on X* . Then X g(x) ± 0 is a unit of for a l l x e X . So the C(X). Since i s hyperreal . This proves that Suppose X has the G^ Z(g) e Z(8X) X g(p) = 0 , g e M* P . i s real compact . is a locally compact and realcompact space, property. Proof : By remark 3.2, i t suffices to prove that every nonempty zero set Z i n X* has nonempty interior. X i s open in 8X. BX i s normal. So Since i s closed. X i s locally compact, by [2, 6.9(d)], So X* Since X* i s C*-embedded i n BX BX i s compact and Hausdorff, by [2, 3D]. Therefore Z = Z(f) fl X* function Define for some g £ C(8X) f e C(8X). such that g(p) =0 h = | f| + | g| , then set i n X Let p e Z. By 3.5, there exists a but g(x) ? 0 Z f\ X* . Now let {x^} p e Z(h) C converging to p . By continuity of h, {hCx^)} h(p) = 0 . Obviously we can choose a subsequence {x a of {x } a for a l l x e X . such that neighbourhood be a converges to } of distinct points i b.(x ) — > 0 . By induction, choose disjoint compact i of x^ _ such that By complete regularity of X, |h(x) - h ^ ) ] there exists a function < -j- for x e w^ such that . OO 0 < < 1 , w^x,^ ) = 1 , w [X — V ] = 0 . Let • . defined provided that {x } has no limit point i n i cannot has a limit point i n X by the fact that h ± ± 1 0 of X . Note that w(x ) = 1 for each H i and w(x)"= 0 1 Now suppose of w^(q) ^ 0 w = £ w , w i s well 1 = 1 . X ; but i n fact, {X- } i is not zero at any point ± OO for x e X — U i=l v i • for some q e X* , we see that every neighbourhood q meets infinitely many V^'s . Thus X* — (ZCw ) n X*) CZ Z(h) . Since< gX h(q) = 0 . This proves that {x., } has a limit i point q i n 8X . As proved already q e X . Thus there exists a subsequence {x } of {x„ } such that w (x„ ) —> w (q). But w3 (x„ ) = 1 for a l l % in 8 B i s compact, 6 1 n, i t follows that Z(h) C w (q) = 1 . So X* - (Z(w^) H X*) ^ <j> . s Z and X* — (Z(w$) H X*) Theorem (3.7) If X Since i s open, this proves the theorem. i s realcompact, then X* has no isolated points. Proof : Suppose p i s an isolated point i n X* . Then there exists a zero set neighbourhood By 3.5, there exists Z(f) of p Z(h) e Z(6X) {p} = Z(f) H Z(h) e Z(6X) . So is disjoint from i n gX X, such that such that {p} Z(f) f\ X* = {p} . p e Z(h) C X* . So is a zero set i n BX . Since by [2, 9.5], {p} contains a copy of N. {p} This leads to a contradiction. Theorem-(3.8) Let X A be a B-subalgebra of Let not compact, then X* X* — 8S*(f) property. |A| = C. has a dense subset of C(X), If, in addition, By 3.7, X* X is 2 A-points. c Obviously, for each i s an open dense subset of X* . By 3.6, X* feA, has the Gg has no isolated points. Now apply [5, 3.2], we see H V is dense i n X* that- C(X) with Let V = { X* — 9S*(f) : f e A } . Proof : of be a locally compact and real compact metric space. | H V\ > 2° . Since and |X*| <_ 2^' = 2° . So | H V\ = 2 C A i s a B-subalgebra . By remark 3.4 (b), O V is precisely the set of A-points of X Theorem (3.9) Let X metric space. Let { A ^ : ^1 with of 2 C Proof : = C be a locally compact and realcompact but not compact aeA} for'each a e A and be a family of 8-subalgebras of C(X) | A | f_ C, then X* has a dense subset points which are simultaneously A -points for a l l a Let V = { X* - 3S* (f) : f e A„ , a e A } . a aeA. Then A H V= n n (X* - 3S^(f)) aeA feAy ^* i s precisely the set of points of X* • that are simlltaneously Appoints for a l l a e A . Applying [5, 3.2] again, H V i s dense i n X* of C(X), |x*| <_ 2 Theorem (3.10) | (\V \ ±_ 2 and = 2 . So C A point in X* C . Since A^ i s a 8-subalgebra | (\ V\ = 2 . C is a C*(X)-point i f and only i f i t i s a P-point of X* . Proof : Since f e C*(X) . So M* = { f e C*(X) : f0 (p) = 0 } , we see that p S .(f)=Z(f6), C* J S* (f) = X* H Z(f3) . Now by definition, a point i n X* A c i s a P-point of X* i f and only i f i t is not an element of the X*-boundary of any zero set of X* , and i s a C*(X)-point i f and only i f p 4 3S* (f) = C* = 3(X* H Z ( f ) ) B for a l l f e C*(X) . Obviously a P-point i s a C*(X)-point. Conversely suppose Zi e Z(X*) such that S H X* = Z x p e Z<i C p i s not a P-point. p. e 3Zj . Let S be a G -set of 6 . By [2, 3.11 (b)], there exists a S . Then Then there exists C*(X)-point. Corollary (3.11) (1) 8N — N has a dense subset of 2^ P-points (2) 3R — R has a dense subset of 2 P-points C such that Z e Z(gX) 2 p e 3 (Z H X*) . This proves that 2 BX p such that i s not a Proof : compact. subset of (1) Obviously Furthermore 2 C N i s locally compact and realcompact but not |c*(N)| = C . Applying C*(N) points. By 3.10, 8N — N 3.8, 8N — N has a dense has a dense subset of 2° P-points. (2) compact. p Since R R i s obviously locally compact and realcompact but not i s separable, has a dense subset of 2 C |c*(R)| = C . Applying 3.8 and 3.10, P-pdints. REMOTE POINTS IN gR In this chapter, we shall turn our attention to the remote points V in the space numbers. gR, the Stone Cech compactification of the space As in [2], we associate with each maximal ideal M R in P of real C(R) the z-ultrafilter A For p e gR, = P we denote by is a neighbourhood of 0 Theorem (4.2) that Lemma (4.3) gR R P P } = { Z e Z(R) : p e CI. Z } . the set of a l l f e C(R) for which Cl ,Z(f) OT i.e. { f e C(R) : p e i n t C l Z ( f ) } g R p e gR 3 R i s said to be a remote point in closure of any discrete subset of gR — R Since gR — R 0 A point i s not in the Proof :. p, = P Definition (4.1) p { Z(f) : f e M has a dense subset of is separable, has a dense subset of 2° if R . C-points . |c(R)| = C . By 3.8, i t is immediate 2° C-points. If Z i s a closed nowhere dense set in a discrete subset D of R gR such that D H Z = <J> , R, D U then there exists Z = C1 D D . Proof : Since Z i s closed, R— Z i s open . As an open set i n R, R — Z i s a union of disjoint open intervals a discrete subset D CI I a a I . For each w I such that the endpoints of I limit points of D„ . Put D = U D„ . Obviously a , choose are the only D H Z = <p and DyZ = CI D. a, Theorem (4.4) (1) p For p e 6R, - K the following are equivalent : i s a remote point i n PR . « (2) A (3) M = 0 . (4) p i s a C-point of BR — R . (5) M (6) 0 P P P P Proof : has no nowhere dense member . P i s a minimal prime ideal . i s prime . (1) => (2) . Suppose that By 4.3, there is a discrete subset Z U D = C1„D , so that K Therefore A D Cl Z c P has a nowhere dense member of R such that Cl D . Hence O T OT3 pK Z (\ D = < } > and p e C l Z C Cl D . O T pK OT1 pK pK p i s not a remote point i n BR . (2) =>(1) . Suppose is a discrete subset D p i s not a remote point i n BR . Then there of R such that p e CI D . Clearly pK We claim is open i n D. P OT( i n t C l D ^ <j> . Then D A . Let q e (int CI D) H D . Since So {q} « D (\ G Cl D e A . pK int CI D = <p . Suppose, on the contrary, that (int CLD) H. D {q} C Z. D i s discrete, q for some open set G i n R. Obviously G H (int-.CLD) . Conversely, l e t r e G H (int Cl D) . Then n n r is either a ppint of D r e D-f\ G . Hence or a limit point of D. If r i s a point of D, r = q . If r i s a limit point of D, then G (\ D contains i n f i n i t e l y many points of D. This contradicts the fact that {q} i s open i n R, But this cannot be true. A D ' < (\ G So this cannot be the case, and {q} = G f\ (int-.Cl^D) . is a singleton set. This proves that then i . e . q i s an isolated point i n R. So we must have the fact that int CI D = <j> . So R R has a nowhere dense member . P (2) =>(3) . Suppose that f eM . Since P C1 (R — Z(f)) R there exists a function A has no nowhere dense member. Let P i s a closed.set i n R, by Urysohn's lemma g e C(R) such that R = Z(f) U Z(g) . We claim Z(g) = C1 (R — Z(f)) . Thus R p 4 Cl _Z(g) . Suppose not, then 0 OT p e C l Z ( f ) n Cl Z(g) . By theorem 1.15, ( 4 ) , p e C l Z ( f ) H Cl Z(g) = 3R pR gR = Cl (Z(f) n Z(g)) = CI. a Z(f) . This proves that or> pK pK K 3 Z(f) nowhere dense member. Cl ,Z(g) DT pK Cl Z(f) pK ot> S^ZCf) e A D pK i s closed , $R — Cl Z(g) ori pR i s open. p e int Cl 00 pK of Dt( z f pK P 0 =M P l„T, ( )- fe 0 . P . For any f e C(R) and Z ( f ) . Thus p then feM , P i s i n the interior of pK i n 3R — R . By remark 3.4, (1), this proves that S*(f) c pR c . Thus has no This proves that p e S*(f) = S (f) H (3R-R) = (CI Z(f)) H ( 3 R - R ) , P P OT) i s a neighbourhood of p . Thus fe 0 A So p 4 Cl„ Z(g) . So ' p e 3R — Cl Z(g) c (3) => (4) . Suppose that whence . Since P K i s nowhere dense, this contradicts our hypothesis that R Since gR p i s a C-point 3R - R . (4) => (2) . Suppose that Z eA P . We shall show that p i s a C-point of 3R — R , and let Z i s not nowhere dense. Since Ze A , p .. p e (Cl Z) n p e C l Z 1 So p R p gR (BR - R) « S ( f ) f\ (BR — R) = S*(f) . i s a C-point, by remark 3.4, (1), p BR — R . Thus a subset of i s i n the interior of p e int Cl . Z . Obviously 01J / pK Since c S*(f) i n ( l n t C l Z ) (V R / <p and i s T) or) pK OT> p K p K Z . This proves that Z i s not nowhere dense. (2) => (5) . Assume (2) . Suppose, on the contrary, that is a nonminimal prime ideal. in and Thus M P Let P Z 4 Z[I] , i t follows that contradicts our hypothesis. P So - Z[I] . Since P . Obviously M 3 Z R = 0 P C1(R - Z) e M . P i s nowhere dense. . Since P P C1(R - Z) This i s a minimal prime ideal. P of a l l the prime ideals contained i n M M R = Z U C1(R - Z) e Z.[I] . So (5) => (3) . Assume (5). By [2, 2.8], ideal, i t follows that P I be a prime ideal properly contained . Choose Z e Z[M ] - Z[I] = A 3 Z = Z H C1(R — Z) e M K M 0 i s the intersection P M P i s a minimal prime . (3) => (6) . Obvious . (6) => (5) . Suppose M i s not a minimal prime ideal. P (5) and (2) are equivalent, i t follows that , D Z . Choose disjoint discrete subsets i = 1, 2, where F we see that i has a nowhere dense member P 2 of denotes the derived set of D_! i = 1, 2 . Obviously z-filter A C l ( G - Z) e A R containing F and 2 F^ incomparable . Since ideal contained in M P i G ± but not p R such that Z < and Z [F^] P , i = 1, 2 . By [2, 7.5], 0 P i s not prime. < Z* ^] - Cl^D^-, has a prime [Fj] i s a prime z - f i l t e r i n A , i = 1, 2 . By 1.23 , we see that p for i = 1, 2 . Since are incomparable. Thus D | = Z , i n R . Let G,- = D-J . By [4, 4.2], A Z, Since Gj n G Z [F ] 2 are < — 2 i s a prime contains 0 = Z, P , Theorem (4.5) gR — R has a dense subset of 2 remote points i n gR . Follows immediately from 4.2 and 4.4 . Proof : Theorem (4.6) gR — R remote points in BR Proof : has a dense subset of 2 points which are simultaneously C and P-points of BR — R . Apply 3.9 to the family { C(R), C*(R) } C(R) . Then BR — R has a dense subset df 2° of B-subalgebras of points which are simultaneously C*-points and C-points of gR — R ..By 3.10, C*-points of gR — R are precisely the P-points of gR — R . By 4 . 4 , C-points of gR — R are precisely the remotes points i n gR . Theorem (4.7) of gR — R gR — R has a dense subset of 2 points which are P-points C but not remote points i n gR . Proof : Let V be a closed neighbourhood i n gR Obviously V (\ R i s not pseudocompact. Since of any point i n gR — R . V H R i s closed, by [2, 1.18], i t i s C-embedded i n R . Thus by [2, 1.20], V fl R a copy i s C*-embedded in D of R , by 1.16, N which i s C-embedded i n R . Since gD = g R cl D • Since V is closed i n gR , we see that D* = gD - D = C l D - D C V H R* • Since gR 8N — N , by 3.11, gD - D has we see that a point i n gD — D i s a P-point of gR — R , that 2 C D contains gD - D P-points of i s homeomorphic with gD - D . By [2, 9 M.2] , i s a P-point of gD — D gD — D has 2° i f and only i f i t P-points of gR — R. Since D i s discrete, no point of BD — D i s a remote point of BR . Since V is arbitrary, this proves the theorem. Definition (4.8) in X A space X i s said to be an F-space i f every cozero set i s C*-embedded i n X . Remark (4.9) By [2, 14.27], 8N — N i s a compact F-space and so i s BR - R . Lemma (4.10) Proof : Every i n f i n i t e compact F-space has at least Let X be an i n f i n i t e compact F-space. there i s a countable discrete subset D i s C*-embedded i n X . So D={d : n CI D = BD Since 2 X neN}. non P-points. C i s infinite , By [2, 14 N.5], by 1.16 . Let f e C*(X) be such •A. that f(d ) = n , n £ N . Then for any p e D* = BD — D = C1 D - D , - 1 n X p e Z(f) , but obviously Z(f) i s not a neighbourhood of p . Thus p is not a P-point . Since |BD — D|,= 2 Theorem (4.11) BR — R has a dense subset of 2° points which are neither remote points i n BR Proof : Let V nor P-points of BR R• — be a closed neighbourhood i n BR As i n the proof of 4.7, V fl R* By remark 4.9, , this proves the lemma . C D* contains a copy i s a compact F-space. BR — R • Since D of BR * This proves the theorem . D* = BD — D By 4.10, D* non P-points of D* . So by [2, 9M.2], D* of of any point i n BR — R • has at least i s discrete, no point of BD — D of 8N — N . has at least 2 C 2° non P-points i s a remote point Theorem (4.12) points i n BR Proof : BR — R has a dense subset of 2 points which are remote C but not P-points of BR — R . Let V of any point i n BR be a closed neighbourhood i n BR — By [5, 5.5], there exists an infinite compact set A BR such that A C V ft (BR — R) . Since BR - R R of remote points i n i s an F-space by 4.9, the C*-embedded subset A has By [2, 4L.2], each of these points i s a non P-point of 2 C non P-poirits. i s also an F-space by [2, 14.26]. BR — R . This proves the theorem . By 4.10, A ... . . . . „ _. •• . - 28 ! • ': CHAPTER V PRIME IDEAL STRUCTURE AND REMOTE POINTS Definition (5.1) Let P(X) denote the family of a l l prime z-filters on X. A prime z - f i l t e r i s said to be minimal i f i t i s a minimal element of P(X). For A, 8 e P(X), i f A C of and that B B 8 , we say that A i s a predecessor i s a successor of A . If i n addition there is no prime z - f i l t e r between them, we use the term immediate predecessor and immediate successor. Theorem (5.2) Z eA Let A be a prime z - f i l t e r on X . Suppose there exists such that for any zero set W ^ A , ZyW^X. Then A i s non- minimal . Proof : For any E q X,, let z(E) = { Z e Z(X) : E C Z } , By assumption, we have z(X — Z) c B . A . Now l e t -'{We Z(X) : z(W - Z) c A } Since X e B , B f <|> . Furthermore B has the following properties : (i) 8 that i s closed under supersets : Let We 8 W C V . Obviously z(V - Z) C z(W - Z) and l e t V e Z(X) such and hence z(V - Z) C A . —-iilMi, Ve B . Thus (ii) for any W , ^ e Z(X) , i f W 4 B choose V i e z(W - Z) — A x for i = 1, 2, t On the other hand, i t i s obvious that 2 Vj (J V E z(Wj y W — Z) 2 2 maximal among the z-filters contained i n B . Note that z(W - Z) C A , so that F ^ A . Finally we shall prove that Z 2 F, of Setting u there is W-j; e F x H W 2 such that , obviously F i s prime. which i s Z 4 F . Furthermore, W e A . Thus Let Z l F C A , , Zg e Z(X) for i = 1, 2 . By the maximality ^ 9 n H Z 4 B , for i = 1, 2 . ± (Zj U W r\ (Zj u Z^) 4 B . Thus £ F . Since B i s W f l (Zj u leads to a contradiction . Thus we must have that F and by W ft Z± 4 B , i = 1, 2 . By property ( i i ) of 8 , closed under supersets, we see that F Z± 4 f Z2 e F . Suppose W = W 2 2 Now applying Zorn's lemma, there exists a z - f i l t e r with 1 (j W ^ B -. definition of 8 , W for any W e F , We V U V ^ A . for i F .1, 2 . Since A i s prime, i u *W B: then F Z ) 4 F , and this 2 i s prime, and hence i s an immediate predecessor of A . So A i s non-minimal . Theorem (5.3) in M* P Proof : For each i s comparable with M P Obviously M p a minimal prime ideal that p e 3X , every prime ideal J C M P C J P of C*(X) contained H C* . f\ C* i s a prime ideal contained i n C* . Choose such that J C P . By 1.24, it. suffices to show C* . To show this, we f i r s t pass to the ring means of the canonical isomorphism f —> of C*(X) onto C(8X) by C(8X) , and then we pass to the family of prime z-filters on BX . Since C(pX) = { f e C(X) : p e C l ^ ^ C f ) } , the prime ideal i n M p f\ C* i s given by corresponding to M P (M f l C*) = { ge C<gX) : p e C l ^ Z ^ X ) } P we claim (M fl C*) P Z ( f ) = Zg (g) g x 6 B i s a z-ideal. ft C*) ) , then P B ov pA pA for some x Let Z f f ) e Z ( ( M g e C(gX) . Hence Z (f| X) = Z ( f ) fl X = x g x = g (g) H X = Z (g|x) , whence p e Cl^ Z (f|x) . This proves that Z X x f e (M fl. C*) P 6 x and hence (M (\ C*) P 6 the corresponding prime z - f i l t e r on BX K P ={ Z x i s a z-ideal. by K P Now l e t us denote ; obviously Z(8X) : p e C l fZ H X) } E 0 PA Also by [2,.14.7], the minimal prime ideal J i s a z-ideal ; let B , i t suffices to show that P of C(gX) corresponding to denote the corresponding minimal prime z - f i l t e r on BX . Now we are going to show that Z e K J$ BC K P . Let Z e B . To show that p e C I ( Z fl X) . Now l e t V be any BX zero set neighbourhood of p . By [2, 7.15], Since B in such that B V e B and hence V fl Z e B . i s minimal, applying theorem 5.2, we can choose a zero set W not (V fl Z) y W = BX . If i n t f V fl Z) = <{,, then W i s BX dense i n BX and hence W — BX . Thus W e B , but this i s impossible. So we see that int(V fl Z) ^ <J> , and (V (\ Z) fl X ^ $ , whence p e C l ( Z fl X) and Z e K BX o Y P . Thus B C K p and hence J C M f\ C* . P Definition (5.4) If Y C X and p i s a z - f i l t e r on Y, i t i s clear that P # = { Z e Z(X) : Z f \ Y e f } i s a z - f i l t e r on X ; i t i s called the z - f i l t e r induced on X If Y C X and F is called the trace of F Definition (5.5) i s a z - f i l t e r on X , then by p F|Y = {ZftY : ZeF} on Y . A z-ideal i n C* i s an ideal I that contains any function that belongs to the same maximal ideals as some function i n I . Theorem (5.6) on X on Y . If Y i s C*-embedded i n X and F such that every member of F meets Y, Proof : It i s clear that F|Y Z, We at least one of them belongs to F | Y . Since S, T e Z(X) such that prime and F C (F|Y)^ # F|Y , i t follows that Y Z(Y) with Z u W = Y , i s C*-embedded i n X, we (F|Y)^ i s prime. (F|Y) # F is Since we see that . Thus at least one of S, T belongs to whence at least one of Z, W i s a prime z - f i l t e r Z = S ft Y , W = T ft Y . Since (S u T) ft Y = Z \j W = Y , by definition of S U T e (F|Y) then i s a z - f i l t e r on Y . To show that F [ Y is prime, i t suffices to show that for any can choose i s a prime z - f i l t e r (F|Y)* , and belongs to F|Y . Hence F|Y i s prime. Review (5.7) only. In the rest.of this chapter, we consider the real line R By the Stone-(5ech compactification theorem and [2, 2.12], we see that the prime z-ideals contained i n M* P with the prime z-filters on 8X are i n order preserving correspondence contained i n A , by means of the P pK mapping P —> Z[P ] . Under this mapping (\ C* —> K M 6 p (see theorem 5.3), p where K Since set i n 6X of C(R) = p R { Z e Z(pR) : p C1_(Z e R) } f\ i s locally, compact, i t follows that gR — R i s a zero and i s C*-embedded i n PR . Obviously .there i s a bounded unit that belongs to M* p e PR — R . Thus for every P (\ C* ± M P M* P i f and only i f p e PR — R . Theorem (5.8) For any contained in K z-filters on R Proof : of P meets z - f i l t e r on contained in A P P gR . be a prime z - f i l t e r contained i n R . R. the family of prime z-filters on is i n one-to-one corresponding with the family of prime p Let p e pR , By theorem 5.6, Since a prime z - f i l t e r on P C R K , P P| R = { Z A R : i t follows that contained i n A P , K P , then every member Z e P } p| R C A P i s a prime . If 8 is obviously the induced prime z-filter 8 is contained i n P C K P K p # = and { Z e Z(pR) : B^|x = 8. Z H R e 8 } Hence the mapping i s onto the family of prime z-filters of C(R) P —> p|x for contained in A P . P - (P|x)^ To prove that the mapping i s one to one, i t suffices to show that Obviously ? c (P|R) such that Z H R = W (\ R . Obviously • Conversely for any # W C Z U ' (6R - R) e P . By definition of P i s prime, we have K Z e (P|R)* , there i s We Z U (6R - R) , , we see that P Z e P . This proves that (PJR)^ C f so that BR - R 4 P . P Since and hence P = (P|R) . # Corollary (5.9) M P fl C* The family of prime z-ideals of contained in i s order isomorphic with the family of prime z-ideals of contained i n M P Proof : C*(R) C(R) . It follows immediately from 5.8, the Stone-c'ech compactif ication theorem and [2, 2.12] . Corollary (5.10) M P fV C* M i s a minimal prime ideal of P i s a minimal prime ideal of Corollary (5.11) p pR For any properly containing of prime z-filters on i f and only i f M P (\ C* C* . p e BR K i f and only i f . i s a remote point in BR is a minimal prime ideal of Theorem (5.12) C C p BR — R — R The family of prime z-filters on i s in one-to-one correspondence with the family contained i n A _ P p K — K Proof : Let P be a prime z - f i l t e r on BR Obviously every member of P B # P R _ . Let B R R = { Z e Z(BR) : Z i t follows from theorem 5.3 that be a prime z - f i l t e r on BR - R 4 K B^' properly contains P —> P | (BR — R) , for K C P p prime z-filters on BR — R contained i n A ? _ K P and g R - R e P . This proves exists We P (P| (BR - R ) ) such that Z O (BR - R) = W H BR — R e P . Suppose not, then the z-ideal P contains no unit of C(R) . Let f e P neighbourhood of p P = (P| (BR — R))^ . . Now let Z e (P| (BR - R)) # i n BR • Since P^ P # i s onto the family of to show that i t i s one-to-one . It suffices to show that P C B, • . Finally we are going R Obviously A^, ($R — R) e B } f\ is clearly prime and B*| (BR - R) = B . Since that the mapping pC . The induced z - f i l t e r > r p i s a prime z - f i l t e r on BR ~ R • Since p|(BR-R)CA contained i n A | _ BR — R K . meets BR — R . So by theroem 5.6, we see that the trace P | (3 R — R) i t follows that properly containing , then there # (BR - R) . We claim i n C*(R) corresponding to and l e t V be a zero set i s prime and i s contained i n A P by [2, 41.4] , i t follows that Ve ZtP ] . Thus 6 hence V (\ Z(f) e ZfP] . Since Hence p e CI Z(f) and therefore B R . i.e. P i s contained i n K BR — R e P * Thus proves that P P V H Z ( f ) e ZfP ] 6 , P R and 8 contains no unit of C(R), V fl Z(f) / $ . f e-M P . This proves that , but this i s impossible. Z f\ (BR - R) = W H (BR - R) e P P C M p f\ C* , So we.must have and hence Z e P . This (p| (BR — R))^C P , and hence the mapping i s one to one . Definition (5.13) Z f The z - f i l t e r generated by a z - f i l t e r that meets every member of F (F, Z) . i s denoted by and a zero set Obviously (F, Z) = { W e Z(X) : for some F e F, F f\ Z ci W }. Remark (5.14) any In the last part of the proof of 5.12, we showed that for p e 6R — R , a prime z - f i l t e r contained i n A i f and only i f i t contains the zero set 6R R . This means that — an immediate successor generated by K P properly contains p K P K p has ( K ) i n the family of prime z-filters on BR , p + and the zero set BR — R . i - e . ( K ) = (K , BR — R) • P + P Furthermore, according to the construction of the one to one onto mapping i n theorem 5.12, we note that ( K ) = (Ztof^ P # D R BR Theorem (5.15) = (K ) P + (Z[0 1 P , BR - R) = ( Z [ 0 R _ ]) P R P z-ideals of C*(R) consists of a l l functions Proof : . Hence (\ C* f R such that f^ vanishes on For any Z e (Z[0^ ], B R - R ) , there exists W e Z[0^_] D pK W H (BR ~ R) C Z . Since Z H ( B R - R ) e Z[0 P pK for any Z e (Z[0 P pK that P i n the family of prime K W f\ ( B R — R ) £ Z[0 Z fl ( B R — R ) So there is _] . Thus K ] ) * , then J P Z e (Z[0 P D ]) , i t follows K # . Conversely BR — R Z f\ ( B R — R ) e Z[0 „ P pK — R 1 . This means i s a zero set neighbourhood of p E f R - R W E Z[0 ] such that P B K such pR p K that (Z[0 , BR - R) i n BR — R . — that # R , and the immediate successor of M a neighbourhood of p ]) . + WH (BR — R) C Z H in B R — R . ( B R — R ) . Thus W fl (BR - R) C Z , and Z e (Z[0 ] , BR - R) . p R P Corollary (5.16) and only i f M* P For any p e B R - R , p Is a P-point of BR — R i f i s the immediate successor of M fl G* P i n the family of prime z-ideals of C*(X) . Corollary (5.17) C*(R) For any p e BR — R , the family of prime z-ideals of contained i n M* p i f and only i f p Theorem (5.18) consists of just the two ideals i s both a remote point i n gR p i s a remote point i n gR contained i n M form a chain . Proof : i s a remote point i n gR-, P If p M* and M p p fl C* and a P-point of gR — R . i f and only i f the prime ideals then M i s a minimal prime p ideal and hence the necessity follows immediately. Conversely, suppose that the prime ideals contained i n M P a chain of C . By [2, 2.8] 0 = fl C . To show that P gR — R , i t suffices to show that a 4 ft C , b 4 C . Then there exists Since P C i s a chain, i t follows that i s prime, ab 4 P • Hence p form i s a remote point 0 = fl C i s prime. Now l e t P, a 4 P , b 4 J • P PC ab 4 ft C J e C such that J, say. Thus b 4 P • . This proves that 0 P Since i s prime. BIBLIOGRAPHY [1] N.J. Fine and L. Gillman, Remote points i n BR, 13 (1962), pp. 29-36 . [2] Proc. Amer. Math. Soc. s . • L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand Princeton, 1960 . [3] Mark Mandelker, Prime ideal structure of rings of bounded continuous functions, Proc. Amer. Math. Soc. 19 (1968), pp. 1432-1438 . [4] , Prime z-ideal structure of C(R), Fund. Math. 63 (1968), pp. 145-166 . [5] Donald Plank, On a class of subalgebras of C(X) with application to BX - X , Fund. Math. 64 (1969), pp. 41-54 .
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Remote points in br and p-points in br - r Leung, Chi-Ming 1971
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Title | Remote points in br and p-points in br - r |
Creator |
Leung, Chi-Ming |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | We are going to study the remote points in βR and the P-points in βR - R. A remote point in βR is a point which is not in the βR chosure of any discrete subset of R. A point p ε βR - R is a P-point of βR - R if every Gδ-set containing p is a neighbourhood of p. |
Subject |
Functions -- Continuous |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080462 |
URI | http://hdl.handle.net/2429/34412 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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