DOCSLIB.ORG

Structure Spaces for Rings of Continuous Functions with Applications to Realcompactifications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Structure Spaces for Rings of Continuous Functions with Applications to Realcompactifications FUNDAMENTA MATHEMATICAE 152 (1997) Structure spaces for rings of continuous functions with applications to realcompactifications by Lothar R e d l i n (Abington, Penn.) and Saleem W a t s o n (Long Beach, Calif.) Abstract. Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone–Cechˇ compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuous functions A(X). 1. Introduction. Let X be a completely regular space and let A(X) be a collection of continuous real-valued functions on X which form a ring under pointwise operations. Two special cases are C(X), the ring of all continuous functions on X, and C∗(X), the ring of bounded continuous functions on X. We study the class of rings of continuous functions which are closed under local bounded inversion (as defined in Section 2). This class includes any ring that contains C∗(X), and any uniformly closed subring of C∗(X), as well as 1 others, including C0 (X), the ring of continuously differentiable functions on a locally compact subset X of R which vanish at infinity (see [1]). Structure spaces for C(X) and C∗(X) have been studied extensively. (See for example [5], where it is shown, by different methods, that the structure space of each ring is isomorphic to βX, the Stone–Cechˇ compactification of X.) We show that for any ring closed under local bounded inversion, the structure space is compact, and is homeomorphic with a quotient of the Stone–Cechˇ compactification βX (Theorem 3.6). Our proofs use a map which assigns a z-filter to every noninvertible f ∈ A(X); this map extends to one from ideals (maximal ideals) to z-filters (z-ultrafilters). For each A(X) we identify a 1991 Mathematics Subject Classification: Primary 54C40; Secondary 46E25. Key words and phrases: ring of continuous functions, maximal ideal, ultrafilter, real- compactification. [151] 152 L. Redlin and S. Watson subspace υAX of M(A) which we call the A-compactification of X. We show that υAX is a realcompactification of X and that every realcompactification arises in this way (Theorem 4.6). Thus every realcompactification of X is a quotient of a subspace of βX. We identify a class of rings which is in natural one-to-one correspondence with the realcompactifications of X (Theorem 4.7). As an application of our results, we prove an extension of the Banach– Stone theorem: Let X and Y be compact and let A(X) and B(Y ) be closed under local bounded inversion; if A(X) and B(Y ) are isomorphic, then X and Y are homeomorphic. (See the remark following Theorem 4.5.) Rings of continuous functions other than C(X) and C∗(X) are also stud- ied in [1], [6], [7], and [8]. 2. Ideals and z-filters. Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X.A zero set in X is a set of the form Z(f) = {x ∈ X : f(x) = 0} for some f ∈ C(X); the complement of a zero set is called a cozero set. We define Z[A(X)] = {Z(f): f ∈ A(X)}; the collection Z[C(X)] of all zero sets is denoted by Z[X]. We always assume that the rings A(X) that we consider contain the constants and separate points and closed sets in X. We make this assumption because of the following easily proved fact: Z[A(X)] is a base for the closed sets in X iff A(X) separates the points and closed sets of X. If f ∈ A(X) and E is a cozero set in X, then f is E-regular if there exists g ∈ A(X) such that fg|E ≡ 1; that is, f is locally invertible on E. To each f ∈ A(X) we attach a collection ZA(f) of subsets of X defined by c ZA(f) = {E ∈ Z[X]: f is E -regular}. Clearly, ZA(fg) ⊂ ZA(f) ∩ ZA(g). It can be shown, as in [9], Theorem 1, that ZA(f) is a z-filterS on X iff f is not invertible in A(X). For S ⊂ A(X) we write ZA[S] = f∈S ZA(f). It was shown in [9] and [3] that if A(X) is a uniformly closed subring that contains or is contained in C∗(X), then for an ideal I in A(X), ZA[I] is a z-filter on X. The proofs there depend on the assumption that A(X) is uniformly closed. In Theorem 2.1 below we show that this is in fact true for any subring of C(X). The inverse of the map ← ZA, considered as a set map, is denoted by ZA and defined by ← ZA [S] = {f ∈ A(X): ZA(f) ⊂ S}, where S is a collection of zero sets in X. It follows immediately from the ← ← definition that ZA [ZA[S]] ⊃ S and ZA[ZA [S]] ⊂ S for all S ⊂ A(X) and S ⊂ Z[X]. For a z-filter F on X we define IA[F] = {f ∈ A(X) : lim fh = 0 for all h ∈ A(X)}, F where limF f denotes the limit of the filter base f(F). Clearly, IA[F] is an ideal of A(X). Rings of continuous functions 153 Theorem 2.1. (a) If I is an ideal in A(X) then ZA[I] is a z-filter on X. ← (b) If F is a z-filter on X then IA[F]⊂ ZA [F]. P r o o f. (a) Clearly, ∅ 6∈ ZA[I], because I contains no invertible elements. If F ∈ Z[X] and F ⊃ E ∈ ZA[I], then F ∈ ZA[I]. Now let E, F ∈ ZA[I], and choose f, g ∈ I locally invertible on Ec and F c respectively. Then there exist h, k ∈ A(X) such that fh|Ec ≡ 1 and gk|F c ≡ 1. Let w = fh + gk − fhgk. Then w ∈ I, and since w|Ec∪F c ≡ 1, it follows that w is locally invertible c c c c c on E ∪ F . Thus (E ∪ F ) = E ∩ F ∈ ZA[I], and so ZA[I] is a z-filter. (b) For f ∈ IA[F] we show that for every E ∈ ZA(f) there exists F ∈ F such that F ⊂ E. If no such F exists, then F ∩ Ec 6= ∅ for all F ∈ F. Since E ∈ ZA(f) there exists h ∈ A(X) such that hf|Ec ≡ 1. But then 1 is a cluster point of {fh(F ): F ∈ F}, contradicting the hypothesis that ← limF fh = 0. Thus ZA(f) ⊂ F; that is, f ∈ ZA [F]. ← If F is a z-filter or even a z-ultrafilter on X, then ZA [F] is not necessarily an ideal. For example, in the ring P (R) of polynomials on R, for any z- ← filter F, the set ZA [F] consists of all polynomials other than the nonzero ← constants. We now introduce a class of subrings for which ZA [F] is an ideal for every z-filter F. A subring A(X) of C(X) is closed under local bounded inversion if every element of A(X) that is bounded away from 0 on a cozero set E is locally invertible on E; that is, if f(x) ≥ c > 0 for all x ∈ E, then f is E-regular in A(X). Any subring of C(X) that contains C∗(X) is closed under local bounded inversion, and according to [3], Lemma 1.2(c), so also is any uniformly closed subring of C∗(X). However, a subring of C(X) that is closed under local bounded inversion need not be comparable to C∗(X). (Consider, for example, the ring of all continuous functions f : R → R for which limx→∞ f(x) exists.) In the study of C(X) the zero sets Z(f) play a central role. The following result gives a relationship between the z-filter ZA(f) and the zero set Z(f). PropositionT 2.2. If A(X) is closed under local bounded inversion, then Z(f) = ZA(f). P r o o f. Suppose y 6∈ Z(f) and without loss of generality assume that f(y) > 0. Choose a cozero set neighborhood G of y such that f(x) ≥ c > 0 c for all x ∈TG. By hypothesis f is locally invertibleT on G and so G ∈ ZA(f). Thus y 6∈ ZA(f). This shows that Z(f) ⊃ ZA(f). The other inclusion is immediate. Theorem 2.3. Let A(X) be closed under local bounded inversion. If F ← ← is a z-filter on X then ZA [F] = IA[F]; in particular, ZA [F] is an ideal in A(X). P r o o f. We claim that if f ∈ A(X) and F ⊃ ZA(f), then limF fh = 0 for all h ∈ A(X). To show this, let f be a noninvertible element of A(X). 154 L. Redlin and S. Watson We show that limZA(f) f = 0. Let [−ε, ε] be a neighborhood of 0 in R and −1 let Eε = f ([−ε, ε]). Set F1 = {x ∈ X : f(x) > ε} and F2 = {x ∈ X : f(x) < −ε}. Since A(X) is closed under local bounded inversion, f is F1-regular and F2- c regular, and hence (F1 ∪F2)-regular ([9], Lemma 1(b)).
Load more

Recommended publications
  • On Rings of Sets II. Zero-Sets
    ON RINGS OF SETS H. ZERO-SETS Dedicated to the memory of Hanna Neumann T. P. SPEED (Received 2 May 1972) Communicated by G. B. Preston Introduction In an earlier paper [llj we discussed the problem of when an (m, n)-complete lattice L is isomorphic to an (m, n)-ring of sets. The condition obtained was simply that there should exist sufficiently many prime ideals of a certain kind, and illustrations were given from topology and elsewhere. However, in these illus- trations the prime ideals in question were all principal, and it is desirable to find and study examples where this simplification does not occur. Such an example is the lattice Z(X) of all zero-sets of a topological space X; we refer to Gillman and Jerison [5] for the simple proof that TAX) is a (2, <r)-ring of subsets of X, where we denote aleph-zero by a. Lattices of the form Z(X) have occurred recently in lattice theory in a number of places, see, for example, Mandelker [10] and Cornish [4]. These writers have used such lattices to provide examples which illuminate a number of results concerning annihilators and Stone lattices. We also note that, following Alexandroff, a construction of the Stone-Cech compactification can be given using ultrafilters on Z(X); the more recent Hewitt realcompactification can be done similarly, and these topics are discussed in [5]. A relation between these two streams of development will be given below. In yet another context, Gordon [6], extending some aspects of the work of Lorch [9], introduced the notion of a zero-set space (X,2£).
    [Show full text]
  • Extension of Continuous Functions Into Uniform Spaces Salvadorhernández
    proceedings of the american mathematical society Volume 97, Number 2, June 1986 EXTENSION OF CONTINUOUS FUNCTIONS INTO UNIFORM SPACES SALVADORHERNÁNDEZ ABSTRACT. Let X be a dense subspace of a topological space X, let Y be a uniformizable space, and let /: X —►Y a continuous map. In this paper we study the problem of the existence of a continuous extension of / to the space T. Thus we generalize basic results of Taimanov, Engelking and Blefko- Mrówka on extension of continuous functions. As a consequence, if D is a nest generated intersection ring on X, we obtain a necessary and sufficient condition for the existence of a continuous extension to v(X, D), of a continuous function over X, when the range of the map is a uniformizable space, and we apply this to realcompact spaces. Finally, we suppose each point of T\X has a countable neighbourhood base, and we obtain a generalization of a theorem by Blair, herewith giving a solution to a question proposed by Blair. In the sequel the word space will designate a topological space. By pX we shall denote a uniform space, p being a collection of covers in the sense of Tukey [13] and Isbell [10]. 7/iX will mean the completion of the space pX. Let p be a uniformity on X and let m be an infinite'cardinal number, then pm will denote the uniformity on X generated by the //-covers of power < m (we only consider p and m for which pm is definable; see [6, 10 and 14]). The compact uniform space ipx0X is called the Samuel compactification of pX, denoted by spX.
    [Show full text]
  • Bibliography
    Bibliography AlexandrotT, P.: 1939 Bikompakte Erweiterungen topologischer Raume. Mat. Sbornik 5, 403-423 (1939). (Russian. German summary.) MR 1, p. 318. AlexandrotT, P. and Hopf, H.: 1935 Topologie, Berlin: Springer 1935. AlexandrotT, P., and Urysohn, P.: 1929 Memoire sur les espaces topologiques compacts. Verh. Akad. Wetensch. Amster­ dam 14, 1-96 (1929). Alb,R.A.: 1969 Uniformities and embeddings. Proc. Int. Sympos. on Topology and Its Appls. (Herceg-Novi, 1968), pp. 45-59. Belgrad: Savez Drustava Mat. Fiz, i Astronom. 1969. MR 42 # 3737. 1972 Some Tietze type extension theorems. General Topology and Its Relations to Modern Analysis and Algebra ,III (Proc. Third Prague Topological Sympos., 1971), pp. 23-27, Prague: Academia 1972. Alb, R. A., Imler, L., and Shapiro, H. L.: 1970 P- and z-embedded subspaces. Math. Ann. 188, 13-22 (1970). MR 42# 1062. Alb, R. A. and Sennot, L. : 1971 Extending linear space-valued functions. Math. Ann. 191, 79-86 (1971). MR 43#6877. 1972 Collection wise normality and the extension of functions on product spaces. Fund. Math. 76, 231-243 (1972). Alb, R. A. and Shapiro, H. L.: 1968A A note on compactifications and semi-normal spaces. 1. Austral. Math. Soc. 8, 102-108 (1968). MR 37#3527. 1968B Normal bases and compactifications. Math. Ann. 175, 337-340 (1968). MR 36#3312. 1969A Wallman compact and realcompact spaces. Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967), pp. 9-14. BerlIn: Deut­ scher Verlag Wissensch. 1969. MR 40#872. 1969B ~-realcompactifications and normal bases. 1. Austral. Math. Soc. 9, 489-495 (1969). MR 39#3455.
    [Show full text]
  • Algebraic Properties of Certain Rings of Continuous Functions
    Pacific Journal of Mathematics ALGEBRAIC PROPERTIES OF CERTAIN RINGS OF CONTINUOUS FUNCTIONS LI PI SU Vol. 27, No. 1 January 1968 PACIFIC JOURNAL OF MATHEMATICS Vol. 27, No. 1, 1968 ALGEBRAIC PROPERTIES OF CERTAIN RINGS OF CONTINUOUS FUNCTIONS Li PI SU n f f Let X and Fbe any subsets of E , and (X , dj and (Y , d2) be any metric spaces. Let O(X), 0 ^ m ^ oo, denote the ring r of m-differentiable functions on X, and Lc(X ) be the ring of the functions which are Lipschitzian on each compact subset of X', and L(Xf) be the ring of the bounded Lipschitzian functions onl'. The relations between algebraic properties m of C (X), (resp. Le(X') or L(X) and the topological properties of X (resp. X') are studied. It is proved that if X and Y, (resp. f 1 (X , dί) and (Y , d2)) are m-realcompact, (resp. Lc-real-compact m or compact) then O(X) = C {Y) (resp. Le(X') = LC(Y>) or L(Xf) = L(Y') if and only if X and Y are O-diffeomorphic 1 (resp. (X ', di) and (Y', d2) are Lc or L-homeomorphic). During the last twenty years, the relations between the algebraic properties of Cm(X) and Cm(Y) and the topological properties of X and Y have been investigated by Hewitt [4], Myers [9], Pursell [11], Nakai [10], and Gillman and Jerison [3], where m is a positive integer, zero or infinite. In 1963, Sherbert [12] studied the ring L(X). Recently, Magill, [6] has obtained the algebraic condition relating C(X) and C(Y) (i.
    [Show full text]
  • Characterizing Realcompact Spaces As Limits of Approximate Polyhedral Systems
    Comment.Math.Univ.Carolin. 36,4 (1995)783–793 783 Characterizing realcompact spaces as limits of approximate polyhedral systems Vlasta Matijevic´ Abstract. Realcompact spaces can be characterized as limits of approximate inverse systems of Polish polyhedra. Keywords: approximate inverse system, approximate inverse limit, approximate resolu- tion mod P, realcompact space, Lindel¨of space, Polish space, non-measurable cardinal Classification: 54B25, 54C56, 54D30 1. Introduction In a series of recent papers various classes of spaces were characterized as spaces which admit resolutions (commutative or approximate), consisting of spe- cial classes of polyhedra. By a polyhedron we mean the carrier of a simpli- cial complex, endowed with the CW-topology. For pseudocompact spaces this was achieved in [6], for Lindel¨of and strongly paracompact spaces in [1], for n- dimensional spaces in [8] and [18] and for finitistic spaces in [12]. In this paper we obtain such characterizations for the class of realcompact spaces. In the liter- ature one can already find two characterizations of realcompact spaces in terms of inverse systems. The first one characterizes realcompact spaces as limits of inverse systems of Lindel¨of spaces ([15, Theorem 23]). The second one char- acterizes realcompact spaces as limits of inverse systems of Polish spaces ([3, Proposition 3.2.17]). Recall that a realcompact space can be defined as a space homeomorphic to a closed subspace of a product of copies of the real line R. Since R is topologically complete and topological completeness is preserved under direct products and closed subsets, every realcompact space is topologically complete. A separable space is called Polish if it is completely metrizable.
    [Show full text]
  • On Realcompact Topological Vector Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector RACSAM (2011) 105:39–70 DOI 10.1007/s13398-011-0003-0 SURVEY On realcompact topological vector spaces J. K¸akol · M. López-Pellicer Received: 7 January 2010 / Accepted: 7 May 2010 / Published online: 3 February 2011 © The Author(s) 2011. This article is published with open access at Springerlink.com Abstract This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vec- tor spaces appearing in Functional Analysis, including Fréchet spaces, (LF)-spaces, and their duals, (DF)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (LF)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis. Keywords Angelicity · Baire and (b-) Baire-like · Bornological · Borel set · C∗-embedded · Class G · (DF) space · Distinguished space · Fréchet–Urysohn · k-Space · K -Analytic · (Weakly) Lindelöf (Σ) · Locally convex space · (Σ-)Quasi-Suslin space · (Strongly) realcompact space · (Compact) resolution · Talagrand compact · (Countable) tightness · Trans-separable · Weakly compact · (WCG) space · Web-bounded (compact) Mathematics Subject Classification (2000) 54H05 · 46A04 · 46A50 Dedicated to Professor Manuel Valdivia, excellent professor and mathematical researcher, on the occasion of his 80th birthday. The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no.
    [Show full text]
  • On the Hewitt Realcompactification of a Product Space
    ON THE HEWITT REALCOMPACTIFICATION OF A PRODUCT SPACE BY W. W. COMFORT(') One of the themes of Edwin Hewitt's fundamental and stimulating work [16] is that the g-spaces (now called realcompact spaces) introduced there, although they are not in general compact, enjoy many attributes similar to those possessed by compact spaces; and that the canonical realcompactification vX associated with a given completely regular Hausdorff space X bears much the same relation to the ring C(X) of real-valued continuous functions on X as does the Stone-Cech compactification ß^to the ring C*(X) of bounded elements of C(X). Much of the Gillman-Jerison textbook [12] may be considered to be an amplification of this point. Over and over again the reader is treated to a "/?" theorem and then, some pages later, to its "v" analogue. One of the most elegant "jS" theorems whose "v" analogue remains unproved and even unstated is the following, given by Irving Glicksberg in [13] and reproved later by another method in [11] by Zdenëk Frolik: For infinite spaces X and Y, the relation ß(Xx F)=JSXx^Fholds if and only if Zx Fis pseudocompact. The present paper is an outgrowth of the author's unsuccessful attempt to characterize those pairs of spaces (X, Y) for which v(Xx Y) = vXxvY. It is shown (Theorem 2.4) that, barring the existence of measurable cardinals, the relation holds whenever y is a ¿-space and uXis locally compact; and more generally (Theorem 4.5) that the relation holds whenever the ¿-space Y and the locally compact space vX admit no compact subsets of measurable cardinal.
    [Show full text]
  • The Epimorphic Hull of C(X)
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 105 (2000) 65–88 The epimorphic hull of C.X/ Robert M. Raphael a;1, R. Grant Woods b;∗ a Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H4B 1R6, Canada b Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada Received 29 June 1998; received in revised form 15 December 1998 Abstract The epimorphic hull H.A/ of a commutative semiprime ring A is defined to be the smallest von Neumann regular ring of quotients of A. Let X denote a Tychonoff space. In this paper the structure of H.C.X// is investigated, where C.X/ denotes the ring of continuous real-valued functions with domain X. Spaces X that have a regular ring of quotients of the form C.Y/ are characterized, and a “minimum” such Y is found. Necessary conditions for H.C.X//to equal C.Y/ for some Y are obtained. 2000 Elsevier Science B.V. All rights reserved. Keywords: Epimorphic hull of a ring; C.X/; Rings of quotients AMS classification: Primary 46E25; 54C30, Secondary 54G10; 54G12 1. Introduction If R is a commutative ring with identity, there exists a well-developed notion of a generalized ring of quotients of R (defined in 2.1(b) below). In particular, each such R possesses a “complete ring of quotients” denoted Q.R/ (see 2.1(c) below). Also, if R is semiprime then Q.R/ is regular in the sense of von Neumann (see 2.1(a) below).
    [Show full text]
  • Realcompact Alexandroff Spaces and Regular a Frames
    LINEAR LIBRARY C01 0068 0754 ', 11111 I U 111111111 _/ UNIVERSITYI I OF Ct\PEE TO\tnl DEPARTMENT OF MATHEMATICS REALCOMPACTREALC PACT ALEXANDRALEXANDROFF F SPACES AND REGULARGUL a-FRa-FRAMES ES Town Christopher R. CapeA. Gilmour of University A thesis prepaprepared under thee superv1s1onsupervision of Associate Professor G.C.L. BrUmmer for the degree of Doctor of Phi,losopPhi·losophy in Mathematics. CopyrightCo ight by the UniversityUniversi 6f Cape Town 1981 The University of CareCaDe Tow'lTow'1 hesn2S been c'·1en(';'!en the right to re1lrDdt1c~re,}r;)dl'c"! t'..'~;t:.:~; Ihth ·si::; iriin 1;hole\ihole or In part. Copydf/1tCopyr:!Jht is I!f':di!!"~d byl.y I.hethe author. ,· The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgementTown of the source. The thesis is to be used for private study or non- commercial research purposes only. Cape Published by the University ofof Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University To Ingrid John and BettyBet Town Cape of University ooothingsDoth which are too wonderful for meoooome The way of an eagle in the air o Qo o0 ·---- ACKNOWLEDGEMENTS I thank my thesis supervisor Associate Professor Guillaume Briimmer for his care, generosity and encouragement. lowe much to Professor K.O. Househam and to e~ch member of the Department of Mathematics, most of whom have been my teachers and all of whom have helped make these thesis years most enjoyable.
    [Show full text]
  • The Hewitt Realcompactification of an Orbit Space
    Turkish Journal of Mathematics Turk J Math (2020) 44: 2330 – 2336 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-1910-38 The Hewitt realcompactification of an orbit space Sadık EYİDOĞAN1;∗, Mehmet ONAT1;2 1Department of Mathematics, Faculty of Science,Çukurova University, Adana, Turkey 2Department of Mathematics, Faculty of Science, Sinop University, Sinop, Turkey Received: 10.10.2019 • Accepted/Published Online: 06.10.2020 • Final Version: 16.11.2020 Abstract: In this paper, we show that the statement in the study of Srivastava (1987) holds also for the Hewitt realcompactification. The mentioned statement showed that when the action of a finite topological group onaTychonoff space is given, the Stone-Čech compactification of the orbit space of the action is the orbit space of the Stone-Čech compactification of the space. As an application, we show that Srivastava’s result can be obtained using themain theorem of the present study. Key words: Realcompactification, orbit space, real maximal ideal 1. Introduction Let G be a topological group and X be a Hausdorff topological space. If the continuous map θ : G × X ! X holds the following statements, then θ is called an action of G on X . 1. θ (e; x) = x for all x 2 X , where e is the identity of G. 2. θ (g; θ (h; x)) = θ (gh; x) for all g , h 2 G and x 2 X . The space X , together with a given action θ of G is called a G-space. We shall use notation gx for θ (g; x). The space X=G = fG (x): x 2 Xg endowed with the quotient topology relative to π is called the orbit space of X , where π is the orbit map from X to X=G and G (x) = fgx : g 2 Gg is orbit of x.
    [Show full text]
  • Intersections of Maximal Ideals in Algebras Between C∗(X) and C(X)
    Topology and its Applications 98 (1999) 149–165 Intersections of maximal ideals in algebras between C∗.X/ and C.X/ Jesús M. Domínguez a;1,J.GómezPérezb;∗ a Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, 47005 Valladolid, Spain b Departamento de Matemáticas, Universidad de León, 24071 León, Spain Received 30 September 1997; received in revised form 8 October 1998 Abstract Let C.X/ be the algebra of all real-valued continuous functions on a completely regular space X, ∗ and C .X/ the subalgebra of bounded functions. There is a known correspondence between a certain ∗ class of z-filters on X and proper ideals in C .X/ that leads to theorems quite analogous to those for C.X/. This correspondence has been generalized by Redlin and Watson to any algebra between ∗ C .X/ and C.X/. In the process they have singled out a class of ideals that play a similar geometric role to that of z-ideals in the setting of C.X/. We show that these ideals are just the intersections of ∗ maximal ideals. It is also known that any algebra A between C .X/ and C.X/ is the ring of fractions ∗ of C .X/ with respect to a multiplicatively closed subset. We make use of this representation to characterize the functions that belong to all the free maximal ideals in A. We conclude by applying our characterization to a subalgebra H of C.N/ previously studied by Brooks and Plank. 1999 Elsevier Science B.V. All rights reserved. Keywords: Rings of continuous functions; Rings of fractions; Prime ideal; Maximal ideal; z-ideal; z-filter AMS classification: 54C40; 13B30 Introduction Let C.X/ be the algebra of all real-valued continuous functions on a nonempty completely regular space X,andC∗.X/ the subalgebra of bounded functions.
    [Show full text]
  • Topology Proceedings
    Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 29, No.2, 2005 Pages 577-594 CHARACTERIZATIONS OF NEARLY PSEUDOCOMPACT SPACES AND RELATED SPACES BISWAJIT MITRA∗ AND SUDIP KUMAR ACHARYYA Abstract. Nearly pseudocompact spaces were initiated by Henriksen and Rayburn. Several topological characterizations of this space are found in literature but any algebraic de- scription is hardly noticed. In this paper we have furnished a number of algebraic descriptions of nearly pseudocompact spaces. In the process we have also introduced Hard pseu- docompact spaces, a generalization of pseudocompact spaces and obtained a few topological characterizations of this space. 1. Introduction Through out our paper, X will be assumed to be a Tychonoff space. It is well known that X is pseudocompact if and only if the Stone-Cechˇ compactification βX is identical with the Hewitt realcompactification υX. In 1980, Henriksen and Rayburn [8] gen- eralized the notion of pseudocompactness by defining a space to be nearly pseudocompact when and only when υX\X is dense in βX\X. These two authors have furnished several characterizations of nearly pseudocompact spaces, most of which by using the notion of hard sets, an entity introduced by the second author in [15]. The main result of their paper is that - X is nearly pseudocompact if 2000 Mathematics Subject Classification. Primary 54C40; Secondary 46E25. Key words and phrases. Nearly pseudocompact space, hard set, hard pseudo- compact, continuous functions with hard support, nonzero ring homomorphism.
    [Show full text]