REMOTE POINTS IN PR AND P-POINTS IN 0R - R by CHI-MING LEUNG B.Sc, New Asia 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 in the Department Qf MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r 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 the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f MfcRe^cpt-f C S The U n i v e r s i t y o f B r i t i s h C o l umbia V a n c o u v e r 8, Canada D a t e ApRiL i 4 • 14 7 1 Supervisor : Professor J.V. Whittaker ABSTRACT We are going to study the remote points i n 0R and the P-points in PR — R . A remote point i n gR i s a point which is not i n the gR chosure of any discrete subset of R . A point p e *R - R i s a P-point of $R—R i f every G^-set containing p is 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 final 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 in 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 in C*(X) has an extension to a function f^ in C(BX) . This thesis is devoted to study the papers [1], [3], [4], [5]. In chapter II, we study the class of subalgebras of C(X) called B-subalgebras. With each 3-subalgebra A of C(X), we define A-po'ints in 3X — X . Then we study the A-points in chapter III . In chapter IV, we turn our attention to the remote points in 6R . Finally, we study the prime ideal structure of C(X) . CHAPTER I PRELIMINARIES Throughout this thesis, a l l given spaces are assumed to be completely regular and Hausdorff. C(X) will denote the collection of a l l real-valued continuous functions on X, and C*(X) will denote the subcollection of bounded functions. Under the pointwise operation, C(X) and C*(X) are commutative rings with identity. A l l ideals in C(X) or C*(X), unmodified, will always mean proper ideals. If S is a set, then | s | will denote the cardinality of S. As is standard, let c denote the cardinality 2^ ° of the continuum. Furthermore, we assume the continuum hypothesis (c = V ^ i ). If S C " » then C1XS, int xS, 8XS will denote, respectively, the closure, interior and boundary of S in X. If f is a function, then we let f < — denote the inverse map. Definition (1.1) For f e C(X), Z(f) = f < _ (0) = { x e X : f(x) = 0 } is called a zero set in X while X — Z(f) is called a cozero set in X . The family Z[C(X)] of a l l zero sets in X will be denoted by Z(X) . Remark (1.2) (1) The family Z(X) of a l l zero sets is a base for the closed sets. (2) f is a unit of C(X) i f and only i f Z(f) = <f> (3) Every zero set is a Gg set . Definition (1.3) Two subsets A and B of X are said to be completely separated in X i f there exists a function f e. C*(X) such that 0 <^ f <_ 1. f[A] = {0} , f[B] = {1} . Definition (1.4) A subspace S of X is said to be C-embedded in X i f every function in C(S) can be extended to a function in C(X) . S is C*-embedded in X i f every function in C*(S) can be extend to a function in C*(X) . Definition (1.5) A non-empty family F of Z(X) is called a z-filter on X provided that (a) <j> i F (b) i f Z(f), Z(g) e F , then Z(f) (\ Z(g) e F (c) i f Z(f) e F , Z(g) e Z(X) and Z(f)CZ(g), then Z(g) e F . If in addition, F is not contained in any other z-filter, then . F is called a z-ultrafilter on X . Theorem (1.6) (a) If I is an ideal [resp. maximal ideal] in C(X), then Z[I]={Z(f):fel} is a z-filter [resp. z-ultrafilter] on X . (b) If F is a z-filter [resp. z-ultrafilter] on X, then Z < -[F]={f:Z(f) eF} is an ideal [resp. maximal ideal] in C(X) . Hence the mapping Z is one-one from the set of a l l maximal ideals in C onto the set of a l l z-ultrafilters. Definition (1.7) An ideal I in C(X) is called a z-ideal i f Z(f) e Z[I] implies f e I . Definition (1.8) A z-filter f in X is called a prime z-filter 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) An ideal I in C(X) or C*(X) is said to be fixed i f H Z[I] f <j> -. Otherwise I is said to be free . Theorem (1.10) (a) The fixed maximal ideals in C(X) are precisely the sets Mp = { f e C : f(p) = 0 } (p e X) . The ideals Mp are distinct for distinct p . For each p, C/Mp is isomorphic with the real field R ; in fact, the mapping Mp(f) —> f(p) is the unique isomorphism of C/Mp onto R . (b) The fixed maximal ideals in C*(X) are precisely the sets Mj = ' { f e C* : f(p) = 0 } (p e X) . The ideals Mp are distinct for distinct p . For each p, C /M* is isomorphic with the real field R ; in fact, the mapping M^(f) —> f(p) is the unique isomorphism of C /M? onto R . Definition (1.11) For p e X, let 0 p denote the set of a l l f in C for which Z(f) is a neighbourhood of p . If Mp = Op , then p is called a P-point of X . Remark (1.12) p e X is a P-point of X i f and only i f every containing p is a neighbourhood of p . Remark (1.13) (a) For p e X, Mp is the only maximal ideal (fixed or free) containing Op • (b) If P is a prime ideal in C, and PCMp , then P D O p . Definition (1.14) By a compactification of a space X, we mean a compact space in which X is dense. Theorem (1.15) Every space X has a Stone-Cech compactification BX with the following equivalent properties : (1) (Stone) Every continuous mapping T from X into any compact space Y has a continuous extension T from 3X into Y . (2) (Stone-Cech) Every function f in C*(X) has an extension to a function f& in C(BX) . (3) (Cech) Any two disjoint zero sets in X have disjoint closures in PX . (4) For any two zero sets Zj and Z 2 in X , (5) If X is dense and C*-embedded in T, then X C T C£X . ( 6 ) If X is dense and C*-embedded in T, then gT = gX . Furthermore, gX is unique, in 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 SCX. S is C*-embedded in X i f and only i f C1DVS = gS . PA (2) The mapping f —> f^ is an isomorphism of C*(X) onto C(gX) . Theorem (1.17) The maximal ideals in C*(X) are precisely the sets M*P = { f e C*(X) : f&(p) = 0 } (p e gX) , and they are distinct for distinct p . The maximal ideals in C(X) are precisely the sets MP = { f e C(X) : p e C l ^ Z ^ f ) } ( p e gX) , and they are distinct for distinct p . Definition (1.18) Let M be a maximal ideal of C(X). [resp. C*(X)] . M is said to be a real ideal i f C/M [resp. C*/M] is isomorphic to the real field R . If M is not real, then we call M hyper-real . Definition (1.19) (a) X is said to be realcompact i f every real maximal ideal in C(X) is fixed. (b) By a realcompactification of X, we mean a realcompact space in which X is dense. (c) X is said to be pseudocompact i f C(X) = C*(X) . Theorem (1.20) is hyper-real i f and only i f M**5 contains a unit of i C Theorem (1.21) Let vX denote the set of a l l points p e BX such that Mp is real. Then (a) vX is the largest subspace of BX in which X is C-embedded. (b) vX is the smallest realcompact space between X and BX . In particular, X is realcompact i f and only i f X = vX . Theorem (1.22) Every (completely regular) space X has a realcompactification vX, contained in BX, with the following equivalent properties. (1) Every continuous mapping x from X into any realcompact space Y has a continuous extension x° from VX into Y . (Necessarily, T° = ^ IvX, where T is the Stone extension of T into BY.) (2) Every function f in C(X) has an extension to a function f° in C(vX) . (Necessarily f v = f*|vX .) Furthermore, the space vX is 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 onto T that leaves X pointwise fixed. Theorem (1.23) If f e C(X) , and aR denotes the one-point compactification of R, then there is a (unique) continuous function f* : BX —> aR which agrees with f on X . Theorem (1.24) In the ring C(X), and also in C*(X), the prime ideals containing a given prime ideal form a chain. (A chain is a totally ordered sets.) B-SUBALGEBRAS Let A be a commutative ring with an identity. Let F be the set of prime ideals in A . For E c A, define V(E) = { P e F : E C p } Note that (1) V(+) = F (2) V(A) = <f> (3) V( u E±) = H V(E±) ieJt ie£ (4) V(E H F) = V(E) U V(F) E±C A , i e £ , where H is an index set E cr A, F c A . Therefore the V's determine a topology on F . This topology is called the hull-kernel topology. Now for a e A , define and let V(a) = { P e F : a e P } F a = F — V(a) . Theorem (2.1) (1) { F a : a e A } is a basis of open sets for F with tae hull-kernel topology. , (ii) F is compact. Proof : (i) Let 8 be a closed subset in F , then B - V(E) for some E C A . Now P e F - B i f and only i f P 4 B i f and only i f E <£ P i f and only i f there exists a e E such that a P i f and only i f there exists . a e E such that P e F a . Thus F — B = u F a . aeE (ii) Suppose F = U F a , E C A . Let I = (E) = ideal generated aeE by E. We claim I = A . Suppose I ^ A , then by Zorn's lemma I C P for some P e F , then P e F a for some a e E . . Hence a 4 P • But ae E C I C P , contradicting a y P . Therefore we must have I = A . So r • . 1 = J ^ i a i » a^ e E , b^ e A . Now for P e F , since 1 4 P > there exists i , 1 < i < n such that a..- 4 P . It follows that P e F 0 . This - - 1 a i . proves that F = F U • • • U F a a l d n and F is compact . Notation (2.2) Let denote the collection of maximal ideals in A endowed with the hull-kernel topology. Definition (2.3) By a-subalgebra A of C(X), we mean a subalgebra in the usual sense which contains the constant functions. Given a subalgebra A of C(X). Define for each p e BX , i I I = { f e A : (fg)*(p) = 0 for a l l g e A } where f* maps BX into the one point compactification of R as stated in 1.23 . Let G A = ' { : p E 3 X } - . Theorem (2.4) is a prime ideal in A, p e 3X . Proof : Since 0 e and 1 £ M^ ., we see that $ $ and ^ A . Obviously is an ideal in A . To prove that is prime, i t suffices to show that i f f, g e A with f, g £ , then fg 4 . Now let f, g e A , choose h, k e A such that (fh)*(p) f 0 and (gk)*(p) ± 0 . Then (fghk)*(p) i 0 . Thus fg M J . ' A Definition (2.5) Let x A : BX—> G^ be such that T A(p)'= . A is said to be a B-subalgebra of C(X) i f T a is a homeomorphism of gX onto M A. Remark (2.6) C*(X) and C(X) are 3-subalgebras of C(X) . For f e A , let S A(f) = T a ~ { p e G A : f e P } = ' { p e PX : f e } = n z((f g)*) . geA Since Z((fg)*) is closed in 8X, S A(f) is closed in 6X . Note that is continuous, since { { p e : f e P } : f e A } is a base for the closed sets in G. A . Definition (2.7) A subalgebra A of C(X) is said to be B-determining i f { Z(f*) : f e A } forms a base for the closed sets in 6X . A is said to be closed under bounded inversion i f f is a unit of A whenever f e A with f >_ 1 . Definition (2.8) An ideal I in A is said to be absolutely convex i f f e I whenever f e A and g e l satisfying | f | [ g{ For convenience, we shall abbreviate M. , MP , G , T. and A A A A to M , MP , G , T and S , respectively. Theorem (2.9) Given a subalgebra A of C(X) , the following are equivalent. (1) A is B-determining (2) G is Hausdorff and x is one-to-one (3) T is a homeomorphism Proof : (1) implies (2) . Suppose A is B-determining and let p,q e B x with p i q . By [2, 6.5(b)], there exists Z1 , Zj, e Z(X) such that Zj U Z2 = X and p 4 Clg^Z} , q gf C1^Z2 . Since A is B-determining, { Z(f*) : f e A } is a base for the closed sets in gX . So we can choose f, g e A such that p 4 Z(f*) Z> CI. „Z, and q ^ Z(g*) 3> CI. Z9 . By the pA p A. choice above, -f ^ MP . Thus MP e G — { Ms e G : f e Ms } which is an O S s open set in G . Similarly MH e G — { M e G : g e M } which is an open set in G . Furthermore by the choice of f, g, we see that fg = 0 . Thus { MS e G : f' e_ M8 } U { Ms e G : g e Ms } = G . So G — { Ms e G: f e Ms } and G — { M S e G : g e M s } are disjoint open sets in G . Since p, q are arbitrary, G is Hausdorff. Since MP ± Mq , x is one-to-one . (2) implies (3) . It suffices to prove that x is closed. Let F be a closed set in BX . Since BX is compact, F is compact . Since x is continuous, x[F] is compact. Since G is Hausdorff, x[F] is closed. (3) implies (1) . Let F be a closed set in BX and p e BX with p 4 F . Since x is a homeomorphism, { S(f) : f e A } is a base for the closed sets in BX . Thus there exists f e A such that p 4 S(f) and F C S(f) . Since S(f) = H Z((fg)*) , (fg)*(p) f 0 for some g e A . geA Thus pV Z(f*) ; but F C S(f) C Z((fg)*) . This proves that {Z(f*): f e A} is a base for the closed sets in BX . Theorem (2.10) Given a subalgebra A of C(X), the following are equivalent. (1) A is closed under bounded inversion. (2) If I is an ideal in A, then f\ Z(f*) f <p . fel (3) Every ideal in A is contained in some Mp . (4) , MA C G A . jtaiiisM (5) Every M e is absolutely convex Proof : (1) implies (2). Let I be an ideal in A . Let F={ Z(f*):& 1} . To prove (2), by the compactness of 8X , i t suffices to show that f has the finite intersection property. Let f 1 , • • •, f n e I and let 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 p e 8X . Since 8X is compact, there exists r > 0 such that |g*(p)| ±_ r > 0. So g >_ r, and g is a unit of A . Since g e l and since I is proper, this is a contradiction. So we must have Z(g*) = cj> . (2) implies (3) . Let I be an ideal in A . Let p e H Z(f*). fel We claim that I C Mp . For i f f e l , then fg e I for a l l g e A . So (fg)*(p) = 0, for a l l g e A . So f e MP . (3) implies (4) . Obvious. (4) implies (5). It suffices to show that MP is absolutely convex. Let f e A and g e MP satisfying | f | <_ |g| . Then | fh| <^ | gh| for a l l h e A . Since X is dense in gX , | (fh)*| <_ | (gh)*| for a l l h e A . So f e Mp . (5) implies (1). Since 1 does not belong to any maximal ideal, i t follows that f is a unit of A whenever f e A with f ^ _ 1 . This completes the proof . Theorem (2.11) Given a subalgebra A of C(X), the following are equivalent. (1) A is a B-subalgebra of C(X) . (2) ' A is B-determining and closed under bounded inversion. Proof : Suppose A is a B-subalgebra of C(X) . By 2.9, A is B-determining. By 2.10, A is closed under bounded inversion. Conversely suppose (2) holds.- By 2.9, T is a homeomorphism of BX onto G . By 2.10, M C G . Since G is T 2 , no two ideals of G are comparable. So M = G . This proves that A is a B-subalgebra of C(X) . THE A-POINTS OF gX - X Let A be a g-subalgebra of C(X). By 2.9, the family {S(f): feA} forms a base for the closed sets in BX. Let X* denote gX — X . For f e A , let S*(f) = S(f) (\ X* . Then { S*(f) : f e A } is a base for the closed sets in X* . For convenience, let us agree that the symbols "CI" , "int" and " 9 " , without subscripts, refer to the topology of X* . Definition (3.1) A space X is said to have the Gg-property i f every nonvoid Gg subset of X has a nonvoid interior. Remark (3.2) Since in a completely regular space X, every Gg containing a compact set S contains a zero set containing S, i t follows that X has the G^-property i f and only i f every nonempty zero set in X has a nonempty interior. The following theorem will be used several times throughout this thesis : Let Y be a nonvoid locally compact Hausdorff space with the Gg-property. If V is a family of at most V^x dense open subsets of Y, then H V is dense in Y. If, in addition, Y has no isolated points, then | n V\ >_ 2*1 . ([5, 3.2]). Definition (3.3) Given a g-subalgebra A of C(X), a point p e X* is said to be an A-point of X* i f , for a l l f e A , p i 3S*(f) . Remark (3.4) (1) A point p e X* is an A-point of X* i f and only i f S*(f) is a neighbourhood of p whenever f e A and p e S*(f) . (2) The set of A-points of X* is precisely f\ (X* — 3S*(f)) . feA Theorem (3.5) X is realcompact i f and only i f for every p e X* , there is a Z e Z(BX) such that p e Z C X* . Proof : Suppose X is realcompact and p e X* . Then MP is hyperreal by [2, 8.4] . By 1.20, M*P contains a'unit f of C(X). Since f is a unit of C(X), i t follows that Z(f^) C X* . By 1.17, p e Z(f3) . This proves the necessity. Conversely, let p e X* . By assumption, there exists Z(g) e Z(8X) such that p e Z(g) C X* . Then g(x) ± 0 for a l l x e X . So the restriction of g on X is a unit of C(X). Since g(p) = 0 , g e M*P . By 1.20, MP is hyperreal . This proves that X is real compact . Theorem (3.6) Suppose X is a locally compact and realcompact space, then X* has the G^ property. Proof : By remark 3.2, i t suffices to prove that every nonempty zero set Z in X* has nonempty interior. Since X is locally compact, by [2, 6.9(d)], X is open in 8X. So X* is closed. Since BX is compact and Hausdorff, BX is normal. So X* is C*-embedded in BX by [2, 3D]. Therefore Z = Z(f) fl X* for some f e C(8X). Let p e Z. By 3.5, there exists a function g £ C(8X) such that g(p) =0 but g(x) ? 0 for a l l x e X . Define h = | f| + | g| , then pe Z(h) C Z f\ X* . Now let {x^ } be a set in X converging to p . By continuity of h, {hCx^)} converges to h(p) = 0 . Obviously we can choose a subsequence {x } of distinct points a i of {xa} such that b.(x ) —> 0 . By induction, choose disjoint compact i neighbourhood of x^ _ such that |h(x) - h ^ ) ] < -j- for x e . By complete regularity of X, there exists a function w^ such that OO 0 < < 1 , w^ x,^ ) = 1 , w±[X — V±] = 0 . Let • w = £ w± , w is well 1 . 1 = 1 . defined provided that {x0 } has no limit point in X ; but in fact, {X- } i i cannot has a limit point in X by the fact that h is not zero at any point OO of X . Note that w(xH ) = 1 for each i and w(x)"= 0 for x e X — U v i • 1 i=l Now suppose w^(q) ^ 0 for some q e X* , we see that every neighbourhood of q meets infinitely many V^'s . Thus h(q) = 0 . This proves that X* — (ZCw8) n X*) CZ Z(h) . Since< gX is compact, {x., } has a limit i point q in 8X . As proved already q e X . Thus there exists a subsequence {x } of {x„ } such that wB(x„ ) —> w6(q). But w3 (x„ ) = 1 for a l l % 1 in n, i t follows that ws (q) = 1 . So X* - (Z(w^) H X*) ^ <j> . Since Z(h) C Z and X* — (Z(w$) H X*) is open, this proves the theorem. Theorem (3.7) If X is realcompact, then X* has no isolated points. Proof : Suppose p is an isolated point in X* . Then there exists a zero set neighbourhood Z(f) of p in gX such that Z(f) f\ X* = {p} . By 3.5, there exists Z(h) e Z(6X) such that p e Z(h) C X* . So {p} = Z(f) H Z(h) e Z(6X) . So {p} is a zero set in BX . Since {p} is disjoint from X, by [2, 9.5], {p} contains a copy of N. This leads to a contradiction. Theorem-(3.8) Let X be a locally compact and real compact metric space. Let A be a B-subalgebra of C(X) with |A| = C. If, in addition, X is not compact, then X* has a dense subset of 2cA-points. Proof : Let V = { X* — 9S*(f) : f e A } . Obviously, for each f e A , X* — 8S*(f) is an open dense subset of X* . By 3.6, X* has the Gg property. By 3.7, X* has no isolated points. Now apply [5, 3.2], we see that- H V is dense in X* and | H V\ > 2° . Since A is a B-subalgebra of C(X), |X*| <_ 2^' = 2° . So | H V\ = 2 C . By remark 3.4 (b), O V is precisely the set of A-points of X Theorem (3.9) Let X be a locally compact and realcompact but not compact metric space. Let {A^: a e A } be a family of 8-subalgebras of C(X) with 1^ = C for'each a e A and |A| f_ C, then X* has a dense subset of 2 C points which are simultaneously Aa-points for a l l a e A . Proof : Let V = { X* - 3S* (f) : f e A„ , a e A } . Then Aa • H V = n n (X* - 3S^(f)) is precisely the set of points of X* that aeA feAy ^* are simlltaneously Appoints for a l l a e A . Applying [5, 3.2] again, H V is dense in X* and | (\V \ ±_ 2 C . Since A^ is a 8-subalgebra of C(X), |x*| <_ 2 = 2 C . So | (\ V\ = 2 C . Theorem (3.10) A point in X* is a C*(X)-point i f and only i f i t is a P-point of X* . Proof : Since M*p = { f e C*(X) : f0 (p) = 0 } , we see that S J.(f)=Z(f6), C* f e C*(X) . So S* A(f) = X* H Z(f3) . Now by definition, a point in X* c is 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 is 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 is a C*(X)-point. Conversely suppose p is not a P-point. Then there exists Zi e Z(X*) such that p. e 3Zj . Let S be a G6-set of BX such that S H X* = Zx . By [2, 3.11 (b)], there exists a Z2 e Z(gX) such that p e Z<i C S . Then p e 3 (Z2 H X*) . This proves that p is not a C*(X)-point. Corollary (3.11) (1) 8N — N has a dense subset of (2) 3R — R has a dense subset of 2^ P-points 2 C P-points Proof : (1) Obviously N is locally compact and realcompact but not compact. Furthermore |c*(N)| = C . Applying 3.8, 8N — N has a dense subset of 2 C C*(N) points. By 3.10, 8N — N has a dense subset of 2° P-points. (2) R is obviously locally compact and realcompact but not compact. Since R is separable, |c*(R)| = C . Applying 3.8 and 3.10, p has a dense subset of 2 C P-pdints. REMOTE POINTS IN gR In this chapter, we shall turn our attention to the remote points V in the space gR, the Stone Cech compactification of the space R of real numbers. As in [2] , we associate with each maximal ideal MP in C(R) the z-ultrafilter A P = { Z(f) : f e MP } = { Z e Z(R) : p e CI. Z } . For p e gR, we denote by 0 P the set of a l l f e C(R) for which Cl O T,Z(f) is a neighbourhood of p, i.e. 0 P = { f e C(R) : p e i n t g R C l 3 R Z ( f ) } Definition (4.1) A point p e gR is said to be a remote point in gR i f p is not in the gR closure of any discrete subset of R . Theorem (4.2) gR — R has a dense subset of 2° C-points . Proof :. Since R is separable, |c(R)| = C . By 3.8, i t is immediate that gR — R has a dense subset of 2° C-points. Lemma (4.3) If Z is a closed nowhere dense set in R, then there exists a discrete subset D of R such that D H Z = <J> , D U Z = C1DD . Proof : Since Z is closed, R — Z is open . As an open set in R, R — Z is a union of disjoint open intervals I w . For each I a , choose a discrete subset Da CI I a such that the endpoints of I are the only limit points of D„ . Put D = U D„ . Obviously D H Z = <p and DyZ = CI D. a, - K Theorem (4.4) For p e 6R, the following are equivalent : (1) p is a remote point in PR . « (2) A P has no nowhere dense member . (3) MP = 0 P . (4) p is a C-point of BR — R . (5) MP is a minimal prime ideal . (6) 0 P is prime . Proof : (1) => (2) . Suppose that A P has a nowhere dense member Z . By 4.3, there is a discrete subset D of R such that Z (\ D = <}> and Z U D = C1„D , so that Cl O TZ c Cl O T 3D . Hence p e Cl O TZ C Cl O T 1D . K p K p K p K p K Therefore p is not a remote point in BR . (2) =>(1) . Suppose p is not a remote point in BR . Then there is a discrete subset D of R such that p e CI D . Clearly ClOT(D e A P . p K p K We claim int CI D = <p . Suppose, on the contrary, that int DCl D ^ <j> . Then (int CLD) H. D A . Let q e (int CI D) H D . Since D is discrete, q is open in D. So {q} « D (\ G for some open set G in R. Obviously {q} C G H (int-.CLD) . Conversely, let r e G H (int nCl nD) . Then r is either a ppint of D or a limit point of D. If r is a point of D, then r e D-f\ G . Hence r = q . If r is a limit point of D, then G (\ D contains infinitely many points of D. This contradicts the fact that D'< (\ G is a singleton set. So this cannot be the case, and {q} = G f\ (int-.Cl^D) . This proves that {q} is open in R, i.e. q is an isolated point in R. But this cannot be true. So we must have the fact that int CI D = <j> . So R R A P has a nowhere dense member . (2) =>(3) . Suppose that A P has no nowhere dense member. Let f e MP . Since C1R(R — Z(f)) is a closed.set in R, by Urysohn's lemma there exists a function g e C(R) such that Z(g) = C10(R — Z(f)) . Thus R R = Z(f) U Z(g) . We claim p 4 ClOT_Z(g) . Suppose not, then p e Cl 3 RZ(f) n Cl p RZ(g) . By theorem 1.15, (4), p e Cl g RZ(f) H Cl g RZ(g) = = Cl o r >(Z(f) n Z(g)) = CI. a Z(f) . This proves that S^ZCf) e A P . Since p K p K K K 3 RZ(f) is nowhere dense, this contradicts our hypothesis that AP has no nowhere dense member. So p 4 Cl„DZ(g) . So ' p e 3R — Cl O T )Z(g) c c l „ T , z ( f ) -p K p R p K Since ClDT,Z(g) is closed , $R — Cl o r iZ(g) is open. This proves that p K p R Cl o t >Z(f) is a neighbourhood of p . Thus f e 0 P . pK (3) => (4) . Suppose that 0 P = MP . For any f e C(R) and p e S*(f) = S c(f) H (3R-R) = (CI Z(f)) H (3R-R), then f e MP , whence f e 0 P . Thus p e int 0 0Cl D t (Z(f) . Thus p is in the interior of p K p K S*(f) in 3R — R . By remark 3.4, (1), this proves that p is a C-point of 3R - R . (4) => (2) . Suppose that p is a C-point of 3R — R , and let Z e A P . We shall show that Z is not nowhere dense. Since Z e Ap , .. p e Cl p RZ 1 So p e (Cl g RZ) n (BR - R) « S c(f) f\ (BR — R) = S*(f) . Since p is a C-point, by remark 3.4, (1), p is in the interior of S*(f) in BR — R . Thus p e int 0 1 JCl /. T )Z . Obviously (lnt o r )Cl O T >Z) (V R / <p and is p K p K p K p K a subset of Z . This proves that Z is not nowhere dense. (2) => (5) . Assume (2) . Suppose, on the contrary, that MP is a nonminimal prime ideal. Let I be a prime ideal properly contained in MP . Choose Z e Z[MP] - Z[I] = A P - Z[I] . Since R = Z U C1(R - Z) and Z 4 Z[I] , i t follows that C1(R - Z) e Z.[I] . So C1(R - Z) e MP . Thus 3 Z = Z H C1(R — Z) e MP . Obviously 3 Z is nowhere dense. This K R contradicts our hypothesis. So MP is a minimal prime ideal. (5) => (3) . Assume (5). By [2, 2.8], 0 P is the intersection of a l l the prime ideals contained in MP . Since MP is a minimal prime ideal, i t follows that MP = 0 P . (3) => (6) . Obvious . (6) => (5) . Suppose MP is not a minimal prime ideal. Since (5) and (2) are equivalent, i t follows that AP has a nowhere dense member Z . Choose disjoint discrete subsets , D 2 of R such that D | = Z , i = 1, 2, where D_! denotes the derived set of D-J in R . Let G,- = Cl^D^-, i = 1, 2 . Obviously Cl R(G i - Z) e Ap . By [4, 4.2], Ap has a prime z-filter F i containing G± but not Z, for i = 1, 2 . Since Gj n G2 = Z, we see that and F 2 are incomparable. Thus Z< [Fj] and Z < — [ F 2 ] are incomparable . Since F^ is a prime z-filter in AP, Z< [F^] is a prime ideal contained in MP , i = 1, 2 . By [2, 7.5], Z * - ^ ] contains 0 P , i = 1, 2 . By 1.23 , we see that 0 P is not prime. Theorem (4.5) gR — R has a dense subset of 2 remote points in gR . Proof : Follows immediately from 4.2 and 4.4 . Theorem (4.6) gR — R has a dense subset of 2 C points which are simultaneously remote points in BR and P-points of BR — R . Proof : Apply 3.9 to the family { C(R), C*(R) } of B-subalgebras of C(R) . Then BR — R has a dense subset df 2° 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 in gR . Theorem (4.7) gR — R has a dense subset of 2 C points which are P-points of gR — R but not remote points in gR . Proof : Let V be a closed neighbourhood in gR of any point in gR — R . Obviously V (\ R is not pseudocompact. Since V H R is closed, by [2, 1.18], i t is C-embedded in R . Thus by [2, 1.20], V fl R contains a copy D of N which is C-embedded in R . Since D is C*-embedded in R , by 1.16, gD = c lgR D • Since V is closed in gR , we see that D* = gD - D = Cl g RD - D C V H R* • Since gD - D is homeomorphic with 8N — N , by 3.11, gD - D has 2 C P-points of gD - D . By [2, 9 M.2] , we see that a point in gD — D is a P-point of gD — D i f and only i f i t is a P-point of gR — R , that gD — D has 2° P-points of gR — R. Since D is discrete, no point of BD — D is a remote point of BR . Since V is arbitrary, this proves the theorem. Definition (4.8) A space X is said to be an F-space i f every cozero set in X is C*-embedded in X . Remark (4.9) By [2, 14.27], 8N — N is a compact F-space and so is BR - R . Lemma (4.10) Every infinite compact F-space has at least 2 C non P-points. Proof : Let X be an infinite compact F-space. Since X is infinite , there is a countable discrete subset D = { d n : n e N } . By [2, 14 N.5], D is C*-embedded in X . So CI D = BD by 1.16 . Let f e C*(X) be such •A. that f(d n) = n - 1 , n £ N . Then for any p e D* = BD — D = C1XD - D , p e Z(f) , but obviously Z(f) is not a neighbourhood of p . Thus p is not a P-point . Since |BD — D|,= 2 C , this proves the lemma . Theorem (4.11) BR — R has a dense subset of 2° points which are neither remote points in BR nor P-points of BR — R • Proof : Let V be a closed neighbourhood in BR of any point in BR — R • As in the proof of 4.7, V fl R* contains a copy D* = BD — D of 8N — N . By remark 4.9, D* is a compact F-space. By 4.10, D* has at least 2 C non P-points of D* . So by [2, 9M.2], D* has at least 2° non P-points of BR — R • Since D is discrete, no point of BD — D is a remote point of BR * This proves the theorem . Theorem (4.12) BR — R has a dense subset of 2 C points which are remote points in BR but not P-points of BR — R . Proof : Let V be a closed neighbourhood in BR of any point in BR — R By [5, 5.5], there exists an infinite compact set A of remote points in BR such that A C V ft (BR — R) . Since BR - R is an F-space by 4.9, the C*-embedded subset A is also an F-space by [2, 14.26]. By 4.10, A has 2 C non P-poirits. By [2, 4L.2], each of these points is a non P-point of BR — R . This proves the theorem . ... . . . .„ _. • • . — - i i l M i , - 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-filter is said to be minimal i f i t is a minimal element of P(X). For A, 8 e P(X), i f A C 8 , we say that A is a predecessor of B and that B is a successor of A . If in addition there is no prime z-filter between them, we use the term immediate predecessor and immediate successor. Theorem (5.2) Let A be a prime z-filter on X . Suppose there exists Z e A such that for any zero set W^A, Z y W ^ X . Then A is 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 A . Now let B -'{We Z(X) : z(W - Z) c A } Since X e B , B f <|> . Furthermore B has the following properties : (i) 8 is closed under supersets : Let We 8 and let V e Z(X) such that W C V . Obviously z(V - Z) C z(W - Z) and hence z(V - Z) C A . Thus V e B . (ii) for any Wx , ^ e Z(X) , i f Wt 4 B for i = 1, 2, then u *W B: choose Vi e z(Wi - Z) — A for i F .1, 2 . Since A is prime, V1 U V2 ^ A . On the other hand, i t is obvious that Vj (J V2 E z(Wj y W2 — Z) and by definition of 8 , W2 (j W2 ^ B -. Now applying Zorn's lemma, there exists a z-filter F which is maximal among the z-filters contained in B . Note that Z 4 F . Furthermore, for any W e F , We z(W - Z) C A , so that W e A . Thus F C A , F ^ A . Finally we shall prove that F is prime. Let Zl , Zg e Z(X) with Z2 u Z2 e F . Suppose Z± 4 f for i = 1, 2 . By the maximality of F , there is W-j; e F such that ^ H Z± 4 B , for i = 1, 2 . Setting W = Wx H W2 , obviously 9 n (Zj U £ F . Since B is closed under supersets, W ft Z± 4 B , i = 1, 2 . By property (ii) of 8 , we see that W r\ (Zj u Z^) 4 B . Thus W f l (Zj u Z2) 4 F , and this leads to a contradiction . Thus we must have that F is prime, and hence F is an immediate predecessor of A . So A is non-minimal . Theorem (5.3) For each p e 3X , every prime ideal P of C*(X) contained in M*P is comparable with MP H C* . Proof : Obviously Mp f\ C* is a prime ideal contained in C* . Choose a minimal prime ideal J such that J C P . By 1.24, it. suffices to show that J C MP C C* . To show this, we first pass to the ring C(8X) by means of the canonical isomorphism f —> of C*(X) onto C(8X) , and then we pass to the family of prime z-filters on BX . Since Mp = { f e C(X) : p e C l ^ ^ C f ) } , the prime ideal in C(pX) corresponding to MP f\ C* is given by (MP fl C*)6 = { ge C<gX) : pe C l ^ Z ^ X ) } we claim (MP fl C*)B is a z-ideal. Let Z f f ) e Z o v((M P ft C*)B) , then pA p A Z g x(f) = Zgx(g) for some g e C(gX) . Hence Z x(f| X) = Z g x(f) fl X = = Zg X(g) H X = Zx(g|x) , whence p e Cl^ xZ x(f|x) . This proves that f e (MP fl. C*)6 and hence (MP (\ C*)6 is a z-ideal. Now let us denote the corresponding prime z-filter on BX by KP ; obviously KP = { Z E Z(8X) : p e Cl 0 fZ H X) } PA Also by [2,.14.7], the minimal prime ideal J$ of C(gX) corresponding to J is a z-ideal ; let B denote the corresponding minimal prime z-filter on BX . Now we are going to show that B C KP . Let Z e B . To show that Z e KP , i t suffices to show that p e C I ( Z fl X) . Now let V be any BX zero set neighbourhood of p . By [2, 7.15], V e B and hence V fl Z e B . Since B is minimal, applying theorem 5.2, we can choose a zero set W not in B such that (V fl Z) y W = BX . If i n t f V fl Z) = <{,, then W is BX dense in BX and hence W — BX . Thus W e B , but this is impossible. So we see that int(V fl Z) ^ <J> , and (V (\ Z) fl X ^ $ , whence p e Cl o Y(Z fl X) and Z e KP . Thus B C Kp and hence J C MP f\ C* . BX Definition (5.4) If Y C X and p is a z-filter on Y, i t is clear that P# = { Z e Z(X) : Z f \ Y e f } is a z-filter on X ; i t is called the z-filter induced on X by p If Y C X and F is a z-filter on X , then F|Y = {ZftY : ZeF} is called the trace of F on Y . Definition (5.5) A z-ideal in C* is an ideal I that contains any function that belongs to the same maximal ideals as some function in I . Theorem (5.6) If Y is C*-embedded in X and F is a prime z-filter on X such that every member of F meets Y, then F|Y is a prime z-filter on Y . Proof : It is clear that F|Y is a z-filter on Y . To show that F [ Y is prime, i t suffices to show that for any Z, We Z(Y) with Z u W = Y , at least one of them belongs to F | Y . Since Y is C*-embedded in X, we can choose S, T e Z(X) such that Z = S ft Y , W = T ft Y . Since F is prime and F C ( F | Y ) ^ , i t follows that (F|Y)^ is prime. Since (S u T) ft Y = Z \j W = Y , by definition of (F|Y)# we see that S U Te (F|Y)# . Thus at least one of S, T belongs to (F|Y)* , and whence at least one of Z, W belongs to F|Y . Hence F|Y is prime. Review (5.7) In the rest.of this chapter, we consider the real line R only. By the Stone-(5ech compactification theorem and [2, 2.12], we see that the prime z-ideals contained in M*P are in order preserving correspondence with the prime z-filters on 8X contained in AP , by means of the pK mapping P —> Z[P6] . Under this mapping Mp (\ C* —> Kp (see theorem 5.3), where Kp = { Z e Z(pR) : p e C1_(Z f\ R) } Since R is locally, compact, i t follows that gR — R is a zero set in 6X and is C*-embedded in PR . Obviously .there is a bounded unit of C(R) that belongs to M*P for every p e PR — R . Thus MP (\ C* ± M*P i f and only i f p e PR — R . Theorem (5.8) For any p e pR , the family of prime z-filters on gR contained in Kp is in one-to-one corresponding with the family of prime z-filters on R contained in AP . Proof : Let P be a prime z-filter contained in KP , then every member of P meets R . By theorem 5.6, P| R = { Z A R : Z e P } is a prime z-filter on R. Since P C KP , i t follows that p| R C A P . If 8 is a prime z-filter on R contained in AP , obviously the induced prime z-filter 8 # = { Z e Z(pR) : Z H R e 8 } is contained in Kp and B^ |x = 8. Hence the mapping P —> p|x for P C KP is onto the family of prime z-filters of C(R) contained in A P . To prove that the mapping is one to one, i t suffices to show that P - (P|x)^ Obviously ? c (P|R)# • Conversely for any Z e (P|R)* , there is We f such that Z H R = W (\ R . Obviously W C Z U (6R - R) , so that Z U ' (6R - R) e P . By definition of KP , we see that BR - R 4 P . Since P is prime, we have Z e P . This proves that (PJR)^ C P and hence P = (P|R)# . Corollary (5.9) The family of prime z-ideals of C*(R) contained in MP fl C* is order isomorphic with the family of prime z-ideals of C(R) contained in MP . Proof : It follows immediately from 5.8, the Stone-c'ech compactif ication theorem and [2, 2.12] . Corollary (5.10) MP is a minimal prime ideal of C i f and only i f MP fV C* is a minimal prime ideal of C . Corollary (5.11) p is a remote point in BR i f and only i f MP (\ C* is a minimal prime ideal of C* . Theorem (5.12) For any p e BR — R The family of prime z-filters on pR properly containing Kp is in one-to-one correspondence with the family of prime z-filters on BR — R contained in A P _ p K — K Proof : Let P be a prime z-filter on BR properly containing Kp . Obviously every member of P meets BR — R . So by theroem 5.6, we see that the trace P | (3 R — R) is a prime z-filter on BR ~ R • Since p C A^, i t follows that p | ( B R - R ) C A P R _ R . Let B be a prime z-filter on BR — R contained in A | > r _ R . The induced z-filter B # = { Z e Z(BR) : Z f\ ($R — R) e B } is clearly prime and B*| (BR - R) = B . Since BR - R 4 KP and gR-Re B#, i t follows from theorem 5.3 that B^ ' properly contains KP . This proves that the mapping P —> P | (BR — R) , for Kp C P is onto the family of prime z-filters on BR — R contained in A? R _ • . Finally we are going to show that i t is one-to-one . It suffices to show that P = (P| (BR — R))^ . Obviously P C (P| (BR - R)) # . Now let Z e (P| (BR - R)) # , then there exists We P such that Z O (BR - R) = W H (BR - R) . We claim BR — R e P . Suppose not, then the z-ideal P in C*(R) corresponding to P contains no unit of C(R) . Let f e P and let V be a zero set neighbourhood of p in BR • Since P^ is prime and is contained in A P , P R by [2, 41.4] , i t follows that Ve ZtP6] . Thus V H Z(f 6) e ZfP 8] and hence V (\ Z(f) e ZfP] . Since P contains no unit of C(R), V fl Z(f) / $ . Hence p e CI Z(f) and therefore f e-MP . This proves that P C Mp f\ C* , B R . i.e. P is contained in KP , but this is impossible. So we.must have BR — R e P * Thus Z f\ (BR - R) = W H (BR - R) e P and hence Z e P . This proves that (p| (BR — R))^C P , and hence the mapping is one to one . Definition (5.13) The z-filter generated by a z-filter f and a zero set Z that meets every member of F is denoted by (F, Z) . Obviously (F, Z) = { W e Z(X) : for some F e F, F f\ Z ci W }. Remark (5.14) In the last part of the proof of 5.12, we showed that for any p e 6R — R , a prime z-filter contained in Ap properly contains Kp i f and only i f i t contains the zero set 6R — R . This means that KP has an immediate successor (K p) + in the family of prime z-filters on BR , generated by KP and the zero set BR — R . i-e. (K P) + = (KP, BR — R) • Furthermore, according to the construction of the one to one onto mapping in theorem 5.12, we note that (K P) + = (Ztof^ D ] ) # . B R R Theorem (5.15) (Z[0 P R1 , BR - R) = (Z[0 P R _ R ] ) # . Hence (Z[0 P R, BR - R) = (K P) + , and the immediate successor of MP (\ C* in the family of prime z-ideals of C*(R) consists of a l l functions f such that f^ vanishes on a neighbourhood of p in BR — R . Proof : For any Z e (Z[0^ D], B R - R ) , there exists W e Z[0^_] such — p K p R that W H ( B R ~ R ) C Z . Since W f\ ( B R — R ) £ Z[0P J , i t follows p K K that Z H ( B R - R ) e Z[0P _] . Thus Ze (Z[0P D ] ) # . Conversely p K K B R — R for any Z e (Z[0P ])* , then Z f\ ( B R — R ) e Z[0P„ 1 . This means p K K p K — R that Z fl ( B R — R ) is a zero set neighbourhood of p E f R - R in B R — R . So there is W E Z[0 P ] such that W H ( B R — R ) C Z H ( B R — R ) . Thus B K W fl (BR - R) C Z , and Z e (Z[0P ] , BR - R) . p R Corollary (5.16) For any p e B R - R , p Is a P-point of BR — R i f and only i f M*P is the immediate successor of MP fl G* in the family of prime z-ideals of C*(X) . Corollary (5.17) For any p e BR — R , the family of prime z-ideals of C*(R) contained in M*p consists of just the two ideals M*p and Mp fl C* i f and only i f p is both a remote point in gR and a P-point of gR — R . Theorem (5.18) p is a remote point in gR i f and only i f the prime ideals contained in MP form a chain . Proof : If p is a remote point in gR-, then Mp is a minimal prime ideal and hence the necessity follows immediately. Conversely, suppose that the prime ideals contained in MP form a chain C . By [2, 2.8] 0 P = fl C . To show that p is a remote point of gR — R , i t suffices to show that 0 P = fl C is prime. Now let a 4 ft C , b 4 C . Then there exists P, J e C such that a 4 P , b 4 J • Since C is a chain, i t follows that P C J, say. Thus b 4 P • Since P is prime, ab 4 P • Hence ab 4 ft C . This proves that 0 P is prime. BIBLIOGRAPHY [1] N.J. Fine and L. Gillman, Remote points in BR, Proc. Amer. Math. Soc. 13 (1962), pp. 29-36 . s . • [2] 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 |
AggregatedSourceRepository | DSpace |
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