UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Remote points in br and p-points in br - r Leung, Chi-Ming 1971

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1971_A8 L48.pdf [ 1.79MB ]
Metadata
JSON: 831-1.0080462.json
JSON-LD: 831-1.0080462-ld.json
RDF/XML (Pretty): 831-1.0080462-rdf.xml
RDF/JSON: 831-1.0080462-rdf.json
Turtle: 831-1.0080462-turtle.txt
N-Triples: 831-1.0080462-rdf-ntriples.txt
Original Record: 831-1.0080462-source.json
Full Text
831-1.0080462-fulltext.txt
Citation
831-1.0080462.ris

Full Text

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 . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080462/manifest

Comment

Related Items