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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. ∗The first author’s research grant is supported by CSIR, Human Resource Development group, New Delhi-110012, India. 577 578 BISWAJIT MITRA AND SUDIP KUMAR ACHARYYA and only if X can be expressed as a union of two subspaces X1 and X2, where X1 is a regular closed almost locally compact subspace of X that is pseudocompact, X2 is a regular closed nowhere locally realcompact subspace of X and intX (X1 T X2)=φ [[8],theorem 3.10]. These spaces were subsequently investigated in some de- tails by John Schommer in [17], who treated nearly pseudocompact spaces without using hard sets and rather using co-zero sets, zero sets and special type of open sets. Schommer had in reality adapted the techniques of Blair and van Douwen’s paper [3] on nearly real- compact spaces to render a new development of nearly pseudocom- pact spaces. In the above mentioned papers, the authors have given several characterizations of nearly pseudocompact spaces, most of which are topological in nature. In the present article, our principal intention is to furnish a number of algebraic descriptions of these spaces via some classes of rings of real valued continuous functions on these spaces and then to give a few topological as well as alge- braic descriptions of hard pseudocompact spaces, a notion which we have introduced as a second generalization of pseudocompact spaces that agree with nearly pseudocompactness precisely on the class of pseudocompact spaces. 2. Algebraic description of nearly pseudocompact spaces The main tool to develop the algebraic description of nearly pseu- docompact spaces is the notion of hard set, introduced by M. Ray- burn in [15]. Definition 2.1. A subset H of X is called hard if it is closed in K ∪ X, where K = clβX(υX\X). The following properties of Hard set are established in [15]. Theorem 2.2. For any space X (1) Every hard subset of X is necessarily closed, and the con- verse is true if and only if X is realcompact. (2) Every compact set is a hard set in X and converse is true if and only if X is nearly pseudocompact. (3) Finite unions, arbitrary intersections, and closed subsets of a hard set are hard. CHARACTERIZATIONS OF NEARLY PSEUDOCOMPACT SPACES 579 (4) Let δX = βX \ (K \ X). Then a closed subset F of X is hard in X if and only if there exist a compact set T in δX with T ∩X = F and X is nearly pseudocompact if and only if X = δX (5) A closed subset F of X is hard in X if and only if there exists a compact set K such that for any open neighborhood V of K there exists a realcompact subset P of X so that F \ V and X \ P can be completely separated. The following characterizations of nearly pseudocompact spaces, which are somewhat topological in nature, are due to Henriksen and Rayburn [Theorem 3.2 and theorems 3.8 [8]]. Theorem 2.3. For a space X, the following statements are equiv- alent. (1) X is nearly pseudocompact (2) Each hard subset of X is compact (3) Every regular hard subset of X is compact (4) Each f ∈ C(X) is bounded on every hard subset of X (5) Each f ∈ C(X) is bounded on every regular hard subset of X (6) Every decreasing sequence of nonempty regular hard sets has non void intersection Proof. (1) ⇔ (2): It follows from the fact that X is nearly pseu- docompact if and only if X = δX and a closed subset H of X is hard if and only if there exists a compact set L in δX such that L ∩ X = H. (2) ⇒ (3): It follows trivially. (3) ⇒ (2): Suppose a hard subset of X is not compact. Then there exists a point p in δX such that p/∈ K. Then we can find an open set V in βX such that p ∈ V ⊆ clβXV ⊆ βX\K. Let W = clβXV ∩X. H = clX(intX W ) is a regular hard set whose closure in βX contains the point p and hence not compact, a contradiction. (2) ⇔ (4): It follows from the fact that if a hard set is not compact, then ∅ =(6 clβXH\H) ⊆ δX ⊆ βX\υX. (3) ⇔ (5): Similar proof as in (2) ⇔ (4). (3) ⇒ (6): It follows trivially. (6) ⇒ (3): It follows from the fact intersection of two regular closed sets contains another nonempty regular closed set. 580 BISWAJIT MITRA AND SUDIP KUMAR ACHARYYA In fact suppose (3) is false. Then there exists a point p in the hyper-real zone such that p lie in the closure of the hard set H which is not compact. Then there exists an 0f 0 in C(X) such that p lie in the βX-closure Zn(f), where Zn(f)={x ∈ X : |f(x)|≥n} and ∩n∈NZn(f)=∅. Note that for each n ∈ N, Zn(f) is a regular closed set. Then for each n ∈ N, there exists a nonempty regu- lar closed set Fn such that Fn ⊆ Zn(f) ∩ H.We can choose Fn in such a way that they are decreasing. Thus Fn’s are non empty decreasing sequence of regular hard sets with void intersection, a contradiction. In the above theorem, C(X) and C∗(X) stand as usual for the rings of real valued continuous functions on X and that of all real valued bounded continuous functions on X respectively. We have introduced here two subrings of C(X) to obtain algebraic descrip- tions of nearly pseudocompact spaces which are in form similar to CK (X) and C∞(X) - two well known subrings of C(X), in fact ∗ subrings of C (X) also, where CK(X) is the family of continuous function with compact support and C∞(X) the family of continu- ous functions which vanish at infinity. Notation 2.4. Let CH (X) be the collection of all those f in C(X) which have hard support in the sense that clX(X\Z(f)) is hard in X, where Z(f)={x ∈ X : f(x)=0} and H∞(X) is the collection of those f in C(X) for which {xX : |f(x)|≥1/n} is hard for each ninN. Since every closed subset of a hard set is hard and the finite union of hard subsets of X are hard, it is clear that CH (X) and H∞(X) are subrings of C(X) with CH (X) ⊆ H∞(X) and CH (X) an ideal of C(X). In a nearly pseudocompact space X, there is no distinction between a hard subset and compact subset and therefore in par- ticular we can write CH (X)=CK (X) and H∞(X)=C∞(X) and ∗ consequently then we can also write CH (X) ⊆ H∞(X) ⊆ C (X). We have shown as the following proposition manifests that, each of these relations by itself is characteristic of nearly pseudocompact spaces. CHARACTERIZATIONS OF NEARLY PSEUDOCOMPACT SPACES 581 Theorem 2.5. The following statements are equivalent for a space X. (1) X is nearly pseudocompact. (2) Ck(X)=CH (X). (3) C∞(X)=H∞(X) ∗ (4) CH (X) ⊆ C (X) ∗ (5) H∞(X) ⊆ C (X) ∗ (6) H∞(X) TC (X)=C∞(X) ∗ (7) CH (X) T C (X)=Ck(X) Proof. (1) ⇒ (2), (1) ⇒ (3), (1) ⇒ (4) and (1) ⇒ (5) are immediate consequences of theorem 2.3. Assume that the condition (1) of theorem 2.5 is false, then from theorem 2.3, there exists an f ≥ 0 in C(X) and a regular hard subset H of X such that f is unbounded on H. Since H = clX(intX H), it follows that f is unbounded on intX H and hence there exists a copy of N={1,2,3...} in intX H with f(n) >n2n for each n ∈ N. Now using the complete regularity of X, we can find out for each n ∈ N,agn ∈ C(X) such that \ { } | | 1 gn(n)=1andgn(X intXH)= 0 .
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