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Maps and F-Normal Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 153 (2006) 2735–2744 www.elsevier.com/locate/topol Maps and f -normal spaces John J. Schommer a,∗, Biswajit Mitra b,1 a Department of Mathematics and Statistics, The University of Tennessee at Martin, Martin, TN 38238, USA b Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road, Kolkata 700019, West Bengal, India Received 17 May 2005; received in revised form 14 November 2005; accepted 15 November 2005 Abstract It has long been known that hyper-real maps preserve realcompactness. In this paper we show that hyper-real maps preserve nearly realcompactness as well. We will also introduce the concepts of ε-perfect maps and f -normal spaces and explore them in a way that mirrors Rayburn’s 1978 study of δ-perfect maps and h-normal spaces. © 2005 Elsevier B.V. All rights reserved. MSC: primary 54C10; secondary 54D60, 54D99 Keywords: Nearly realcompact space; f -normal space; Fast set; Hyper-real map; Perfect map; ε-perfect map; Zenor map; WZ-map 1. Introduction There are three different approaches to the study of nearly realcompact spaces in the literature. Johnson and Man- delker referred to nearly realcompact spaces as η-compact spaces in [12] , characterizing them in terms of ideals of continuous functions. Blair and van Douwen produced quite a few results by characterizing nearly realcompact spaces in terms of their relatively pseudocompact cozero-sets in [2]. In this paper we continue the development of nearly re- alcompact spaces begun in [17], one which parallels Henriksen and Rayburn’s approach to nearly pseudocompact spaces initiated in [8]. We will assume that all spaces are Tychonoff. The basic theories and properties of the Stone–Cechˇ compact- ification βX and the Hewitt realcompactification υX will also be assumed. Furthermore we adopt the notation and terminology of [5] for the terms: zero-sets, cozero-sets, z-ultrafilters, real z-ultrafilters, C-embedded, and C*- embedded. A subspace A of X is said to be z-embedded in X if every zero-set of A is the restriction to A of some zero-set of X. A subspace A of X is said to be relatively pseudocompact in X if and only if every continuous function on X is bounded on A if and only if clβX A ⊆ υX. Hyper-real maps will also play an important role in this paper. We say that a map f : X → Y is hyper-real iff the Stone extension fβ : βX → βY satisfies fβ (βX − υX) ⊆ βY − υY.(Note: By a map we will always mean a continuous surjection.) * Corresponding author. E-mail addresses: [email protected] (J.J. Schommer), [email protected] (B. Mitra). 1 The second author’s research grant is supported by CSIR, Human Resource Development group, New Delhi-110012, India. 0166-8641/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2005.11.006 2736 J.J. Schommer, B. Mitra / Topology and its Applications 153 (2006) 2735–2744 We say that a space X is nearly realcompact iff βX − υX is dense in βX − X. The primary tool for the investi- gations of nearly realcompactness in this paper will be the so-called “fast set”. We say that a subset F of X is fast in X if F is closed in X ∪ JX where JX = clβX(βX − υX). We write J instead of JX when there is no danger of confusion about the ambient space. We will also need to refer frequently to the subspace εX of βX given by εX = βX − (J − X) = X ∪ (βX − J). We will have cause later to refer to a result of Blair and van Douwen concerning another generalization of real- compactness, c-realcompactness. A space is said to be c-realcompactiff for every p ∈ βX − X, there is a decreasing { ∈ } ∈ ⊆ − sequence An: n ω of regular closed sets in βX such that p n∈ω An βX X (cf. [6, 1.1]). Blair and van Douwen have shown that every c-realcompact space is nearly realcompact. Not published in their lifetimes, this result was first published posthumously as [16, 14.3]. We re-present the result here to make its proof more widely available. We begin with some lemmas. { ∈ } Lemma 1.1. [2, 1.4] If G is open in X, then G is relatively pseudocompact in X iff whenever Fn: n ω is ∩ =∅ ∈ =∅ a decreasing sequence of regular closed subsets of X with Fn G for all n ω, then n∈ω Fn . Lemma 1.2. [2, 1.7] X is nearly realcompact iff every relatively pseudocompact cozero-set of X is σ -compact. Theorem 1.3 (Blair and van Douwen). Every c-realcompact space is nearly realcompact. Proof. Let X be c-realcompact and let P = coz f , where f : X →[0, 1]. Suppose P is not σ -compact. By the last = ←[ 1 ] ←[ 1 ] lemma, it suffices to show that P is not relatively pseudocompact. Since P n∈ω f n , 1 and each f n , 1 is = ←[ 1 ] a zero-set, P must contain one non-compact zero-set Z f n , 1 . Now since Z is closed in X and not compact, ∈ − ∩ 0 X must not be compact, so there is a point p (βX X) clβX Z. Since X is c-realcompact, there is a decreasing { ∈ } ∈ ⊆ − ⊆ ∩ =∅ sequence Un: n ω of open sets in X with p n∈ω clβX Un βX X. Since Z P , P clX Un for all ∈ =∅ 2 n ω.But n∈ω clX Un . Thus by Lemma 1.1, P is not relatively pseudocompact. Finally, we will make frequent reference to the results contained in the following theorem [17]: Theorem 1.4. The following are true: (1) F is fast in X iff F = T ∩ X where T is a compact subset of εX. (2) A closed subset F is fast in X iff there exists a compact set T such that for every open set G containing T , F − G is completely separated from the complement of some pseudocompact set in X. (3) Every compact set in X is fast, but every fast set is compact iff X = εX iff X is nearly realcompact. (4) εX = {clβX F : F is fast in X}. (5) Every fast set of X is closed, but every closed set is fast iff X is pseudocompact. (6) A fast set in X is relatively pseudocompact, and a regular fast set (i.e. a regular closed set which is fast in X) is pseudocompact. (7) A closed subset of a fast set in X is fast in X. (8) X is locally pseudocompact iff εX is locally compact. Practically every result in this paper will parallel an analogous result contained in [15]. In many cases, the proofs require only an appropriate modification to the proofs offered in Rayburn’s paper. This does not always work smoothly though, and in some interesting cases more will be required of us. 2. ε-perfect maps Recall that a map f : X → Y is called perfect if it is closed, onto, and f ←(y) is compact for every y ∈ Y . Indeed, ← ⊆ ∈ → it can be shown that a map f is perfect iff fβ (y) X for every y Y , where fβ : βX βY is the Stone extension. → ← ⊆ ∈ Definition 2.1. Amapf : X Y is ε-perfect if fβ (y) εX for every y Y . J.J. Schommer, B. Mitra / Topology and its Applications 153 (2006) 2735–2744 2737 Every perfect map is ε-perfect, but the converse need not be true. If Y is compact but X is merely pseudocompact, then there cannot be a perfect map from X to Y . But every map of X onto Y is ε-perfect, since X pseudocompact implies βX = υX, and consequently εX = βX. It should be noted that ε-perfect is equivalent to perfect whenever X is nearly realcompact. → ←[ ]⊆ Lemma 2.2. A hyper-real map f : X Y is ε-perfect iff fβ εY εX. ⇒ ←[ ]⊆ ←[ − ]⊆ ∈ ←[ − ] Proof. ( ) By hypothesis fβ Y εX, so we need only show that fβ εY Y εX.Letx fβ εY Y .If x ∈ X we are done, so assume x/∈ X. Since εX = X ∪ (βX − JX), this leaves us to prove that x ∈ βX − JX. Observe that ∈ ←[ − ]⊆ ← − = − ← x fβ εY Y fβ (βY JY ) βX fβ (JY ). − ← ⊆ − It remains then to demonstrate that βX fβ (JY ) βX JX. But this follows from the fact that f is hyper-real: fβ (JX) = fβ clβX(βX − υX) ⊆ clβY fβ (βX − υX) ⊆ clβY (βY − υY) = JY . ⊆ ← Thus JX fβ (JY ) and we are done. (⇐) This is immediate, not even requiring that f be hyper-real. 2 Note 2.3. The hyper-real condition of Lemma 2.2 cannot be dropped. The counterexample will be more easily dis- cussed in Note 2.11. Now let us recall that a map f : X → Y is called a Z-map iff f(Z)is closed in Y for all zero-sets Z of X, and f is ← = ← ∈ called a WZ-map iff fβ (y) clβX f (y) for all y Y . It is known that every Z-map is a WZ-map, but the converse is not true [11, 1.2,8.2]. Z-maps are also referred to in the literature as z-closed maps (cf. [20]). → ← = ← ∈ Lemma 2.4. Amapf : X Y is an ε-perfect WZ-map iff fβ (y) clεXf (y) for all y Y . Proof. In general we have ← ⊆ ← ⊆ ← clεX f (y) clβX f (y) fβ (y).
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