Realcompact subspaces

We show that every compact space of large enough size has a realcompact subspace of size κ, for κ?c. We also show that an uncountable realcompact space whose pseudocharacter is at most ω₁, has a realcompact subspaces of size ω₁, thus, by continuum hypothesis, every uncountable realcompact space has realcompact subspace of size ω₁.     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X₁, X₂ such that Int X₁ is locally compact, no points of X₂ has a closed realcompact neighborhood, and $\text{Int(X}{}_1\text{?X}{}_2\text{)=?}$. It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.     

A broader view of the almost Lindelöf property

By first finding necessary and sufficient conditions for the realcompact coreflection, νL, and the regular Lindelöf coreflection, λL, of a completely regular frame L to be isomorphic, we define a frame L to be almost Lindelöf if it is Lindelöf or λL → L is a one-point extension. This agrees with the condition “νL is Lindelöf and L is realcompact or νL is a one-point extension”, which would be a frame version of what are called almost Lindelöf spaces. Thus, the condition “νX is Lindelöf”, which is added in the definition of almost Lindelöf spaces, serves only to compensate for the lack of the regular Lindelöf reflection in Top, and can be dispensed with by concentrating on the frame \({\mathfrak {O}X}\) instead of the space X.     

On c-realcompact spaces

Abstract

The c-realcompact spaces are fully studied and most of the important and well-known properties of realcompact spaces are extended to these spaces. For a zero-dimensional space X, the space υ₀X, which is the counterpart of υX, the Hewitt realcompactification of X, is introduced and studied. It is shown that υ₀X, which is the smallest c-realcompact space between X and β₀X, plays the same role (with respect to C_c(X)) as υX does in the context of C(X). It is proved for strongly zero-dimensional spaces, c-realcompact spaces, realcompact spaces and N-compact spaces coincide. In particular, if X is a strongly zero-dimensional space, then υX = υ₀X. It is obsesrved that a zero-dimensional space X is pseudocompact if and only if C_c(X) = C*_c(X), or equivalently if and only if υ₀X = β₀ X. In particular, a zero-dimensional pseudocompact space is compact if and only if it is c-realcompact. It is shown that Lindelöf spaces, subspaces of the one-point compactification (resp., Lindelöffication) of a discrete space with a nonmeasurable cardinal, are c-realcompact space. If X is a pseudocompact space, it is observed that C(X) = C_c(X) if and only if βX is scattered. Finally, the simplest possible proof (with reasoning) among the known proofs, of the well-known fact that discrete spaces of cardinality less than or equal to that of the continuum are realcompact, is given.     

A Nonstandard characterization of realcompactness

Summary We consider in this note a completely regular space X in a nonstandard framework, and show that this space is realcompact iff the prenearstandard points in a hyperfinite set A *X so that st_X(A)=X are nearstandard. This is achieved by considering a linear programming problem in which the variables are Baire measures on X; a form is found for a near optimizer for an associated linear programming problem based on the set A_pns of prenearstandard points in A. The main result is obtained by noting that X is realcompact iff every Baire measure on X taking values 1 and 0 only is unitary atomic     

A spectral characterization of skeletal maps

We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between ω-spectra with surjective limit projections.     

Groups with T1 primitive ideal spaces

We give a simple necessary and sufficient condition for the group C¹-algebra of a connected locally compact group to have a T₁ primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C¹-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C¹-algebra of a discrete finitely generated solvable group is T₁ if and only if the group is a finite extension of a nilpotent group.     

A counterexample to Wood's conjecture

We prove that if the one-point compactification of a locally compact, noncompact Hausdorff space L is the topological space called pseudoarc, then C₀(L,C) is almost transitive. We also obtain two necessary conditions on a metrizable locally compact Hausdorff space L for C₀(L) being almost transitive.     

Baire sets in topological spaces

Summary In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-ech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrówka in [4]; second, the idea included in the proof of the theorem: Every compact Baire set is aG as given inHalmos' text on measure theory [2; Section 51, theorem D].The author wishes to thank Professor W. W.Comfort for his valuable advice in the preparation of this paper.     

Resolvability vs. almost resolvability

A space X is κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X).Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal κ there is an almost κ²-resolvable but not ω₁-resolvable space of dispersion character κ.     

The three space problem in topological groups

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.     

Locally compact spaces of countable core and Alexandroff compactification

We introduce a new cardinal invariant, core of a space, defined for any locally compact Hausdorff space X and denoted by cor(X). Locally compact spaces of countable core generalize locally compact σ-compact spaces in a way that is slightly exotic, but still quite natural. We show in Section 1 that under a broad range of conditions locally compact spaces of countable core must be σ-compact. In particular, normal locally compact spaces of countable core and realcompact locally compact spaces of countable core are σ-compact. Perfect mappings preserve the class of spaces of countable core in both directions (Section 2). The Alexandroff compactification aX is weakly first countable at the Alexandroff point a if and only if cor(X)=ω (Section 3). Two examples of non-σ-compact locally compact spaces of countable core are discussed in Section 3. We also extend the well-known theorem of Alexandroff and Urysohn on the cardinality of perfectly normal compacta to compacta satisfying a weak version of perfect normality. Several open problems are formulated.     

Realcompactness in maximal and submaximal spaces

We study realcompactness in the classes of submaximal and maximal spaces. It is shown that a normal submaximal space of cardinality less than the first measurable is realcompact. ZFC examples of submaximal not realcompact and maximal not realcompact spaces are constructed. These examples answer questions posed in [O.T. Alas, M. Sanchis, M.G. Tka?enko, V.V. Tkachuk, R.G. Wilson, Irresolvable and submaximal spaces: homogeneity versus σ-discreteness and new ZFC examples, Topology Appl. 107 (3) (2000) 259-273] and generalize some results from [D.P. Baturov, On perfectly normal dense subspaces of products, Topology Appl. 154 (2) (2007) 374-383].     

Inequalities for martingale transforms and related characterizations of Hilbert spaces

Suppose that f is a martingale taking values in a Banach space B and g is its transform by a deterministic sequence of numbers in {−1,1}, such that sup_n‖g_n‖≥1 almost surely. We show that a certain family of Φ-estimates for f holds true if and only B is a Hilbert space.     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

Bases of rearrangement invariant spaces

We prove that if E is a rearrangement-invariant space, then a boundedly complete basis exists in E, if and only if one of the following conditions holds: 1) E is maximal and E ≠ L ₁[0, 1]; 2) a certain (any) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a boundedly complete basis in E. As a corollary, we state the following assertion: Any (certain) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a spanning basis in a separable rearrangement-invariant space E, if and only if the adjoint space E* is separable. We prove that in any separable rearrangement-invariant space E the Haar system either forms an unconditional basis, or a strongly conditional one. The Haar system represents a strongly conditional basis in a separable rearrangement-invariant space, if and only if at least one of the Boyd indices of this space is trivial.     

Ideals associated with realcompactness in pointfree function rings

Abstract

Let RL denote the ring of continuous real-valued functions on a com- pletely regular frame L. The support of an α ∈ RL is the closed quotient ↑(coz α)?. We show that if supports are coz-quotients in L, then the set of functions with realcompact support is an ideal. If L satisfies the stronger condition that supports are C-quotients, then this ideal is the intersection of pure parts of the free maximal ideals of RL. The set of functions whose cozeroes are realcompact is always an ideal, which is free if and only if L is locally realcompact if and only if L is (isomorphic to) an open quotient of υL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only if υL → L is a one-point extension of L.     

Lattice repleteness and some of its applications to topology

We deal with the general concept of lattice repleteness. Specifically, we systematize the study of several important special cases of repleteness, namely, realcompactness, α-completeness, N-compactness, and Borel-completeness; we apply our general results on repleteness to specific lattices in topological spaces, in particular, to analytic spaces; we utilize the concept of $\text{G}$_δ-closure to obtain necessary or sufficient conditions for repleteness (this portion of our work generalizes important theorems of Mrówka on Stone-?echcompactification, of Frolik on realcompact spaces, and of Wenjen on realcompact spaces); finally, we extend the measure representation material of Varadarajan and then we utilize the results to obtain further applications to repleteness.     

Homomorphisms on Lipschitz Spaces

 Denote by the family of all real valued functions on a metric space which satisfy a Lipschitz condition on the compact (bounded) subsets of X. We prove that every homomorphism on is the evaluation at some point of X if and only if X is realcompact (every closed bounded subset of X is compact). (Received 4 November 1998; in revised form 31 May 1999)