The β-space property in monotonically normal spaces and GO-spaces

In this paper we examine the role of the β-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a β-space with a Gδ-diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos.     

On metrizability of compactoid sets in non-archimedean locally convex spaces

In 2003, N. De Grande-De Kimpe, J. Kąkol and C. Perez-Garcia using t-frames and some machinery concerning tensor products proved that compactoid sets in non-archimedean (LM)-spaces (i.e. the inductive limits of a sequence of non-archimedean metrizable locally convex spaces) are metrizable. In this paper we show a similar result for a large class of non-archimedean locally convex space with a £-base, i.e. a decreasing base (U_α)_αN^N of neighbourhoods of zero. This extends the first mentioned result since every non-archimedean (LM)-space has a £-base. We also prove that compactoid sets in non-archimedean (DF)-spaces are metrizable.     

Zero-selectors and GO spaces

We are dealing with Vietoris continuous zero-selectors, i.e., they choose for each non-empty closed set F an isolated point in F. We show that the presence of a continuous zero-selector even on a small class of non-empty closed sets of a space X implies that X is scattered if X is metrizable or non-Archimedean or a P-space. Finally, using continuous zero-selectors, we characterize suborderable spaces which are subspaces of ordinals.     

Metrization of Fpp-spaces

A Hausdorff space each subspace of which is a paracompact p-space is an F_pp-space. A space X is a closed hereditary Baire space if each closed subspace of X is a Baire space. Using a delicate theorem of Z. Balogh it is shown that a first-countable F_pp-space that is a closed hereditary Baire space is metrizable.     

Banach Disks and Marrelledness Properties of Metrizalbe <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript> (Ω, <Emphasis Type="Italic">X</Emphasis>)

Let be a non-empty set and X a metrizable locally convex space. We show that the metrizable locally convex space c₀ (, X) is p-barrelled (totally barrelled) if and only if X is p-barrelled (totally barrelled). Some applications for closed graph theorems are included.     

On hereditarily strong Σ-spaces

In this paper the structure of hereditarily strong Σ-spaces (hsΣ-spaces, for short) is dealt with. The main result asserts that an hsΣ-space is the disjoint union of two σ subspaces one of which is an F_σ, the other a G_δ subset. Examples are given that in many ways, this decomposition cannot be improved. Then we investigate the question when an hsΣ-space is a σ-space. It is shown that a GO-space (or a first countable compactum) is metrizable iff it is an hsΣ-space, thereby proving a conjecture of J. van Wouwe. σ-spaces are characterized as being identical with perfect hsΣ-spaces. The question whether a Lindelöf, first countable hsΣ-space is a σ-space is shown to be independent of set theory. A characterization of hsΣ-spaces with no compact subsets of cardinality >2^ω is given.     

Collectionwise normality and selections into Hilbert spaces

A T₁-space X is countably paracompact and collectionwise normal if and only if every l.s.c. mapping from X into a Hilbert space with closed and convex point-images has a continuous selection. This settles a conjecture posed by M. Choban, V. Gutev and S. Nedev [M. Choban, S. Nedev, Continuous selections for mappings with generalized ordered domain, Math. Balkanica (N.S.) 11 (1-2) (1997) 87-95].     

Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b _f -space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b _f -property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric. Received: June 22, 2000; in final form: May 22, 2001?Published online: June 11, 2002     

Cosmic spaces which are not μ-spaces among function spaces with the topology of pointwise convergence

A space is called a μ-space if it can be embedded in a countable product of paracompact F_σ-metrizable spaces. The following are shown:(1) For a Tychonoff space X, if C_p(X,R) is a μ-space, then X is a countable union of compact metrizable subspaces.(2) For a zero-dimensional space X, C_p(X,2) is a μ-space if and only if X is a countable union of compact metrizable subspaces.In particular, let P be the space of irrational numbers. Then C_p(P,2) is a cosmic space (i.e., a space with a countable network) which is not a μ-space.     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

Set-theoretic constructions of nonshrinking open covers

A family {M_α|α?A} is a shrinking of a cover {O_α|α?A} of a topological space if {M_α|α?A} also covers and M_α?O_α for all α?A.?⁺⁺ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.? implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

Universal free G-spaces

For a compact Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free G-spaces. We show that such a universal free G-space cannot be compact.     

Some generalizations of the concept of a p-space

In this article, some generalizations of the concept of a p-space are introduced and studied. The notion of a source of a space in a larger space and the concepts of partial plumage, s-embedding, p-embedding, p^?-embedding, s-space, and p^?-space are defined and studied in depth (see Theorems 2.6, 2.7, 3.2, 4.3, 4.4, 4.10 and their corollaries). An example of a hereditarily p^?-space which is not a p-space and is a perfect image of a hereditarily p-space is indicated (Example 2.9). Among the main results, we establish that if a paracompact space X is p-embedded in a pseudocompact space as a dense subspace, then X is a p-space (Corollary 4.8), and that if X has a countable network and is p^?-embedded in a pseudocompact space, then X is metrizable (Corollary 4.11). The following problem is posed: is every paracompact Gδ-subspace of a pseudocompact space ?ech-complete?     

On the bounded approximation property in Banach spaces

We prove that the kernel of a quotient operator from an L ₁-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case ? ₁-and Figiel, Johnson and Pe?czyński-case X* separable. Given a Banach space X, we show that if the kernel of a quotient map from some L ₁-space onto X has the BAP, then every kernel of every quotient map from any L ₁-space onto X has the BAP. The dual result for L _∞-spaces also holds: if for some L _∞-space E some quotient E/X has the BAP, then for every L _∞-space E every quotient E/X has the BAP.     

An Isometric Representation of the Dual of {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)}

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2, 3, 8]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure.     

On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trindX?α. W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trIndX?λ. T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dimWX?α where dimW is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dimC of the covering dimension.As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dimCX?α belonging to some absolute Borel class.     

Cauty?s space enhanced

The title refers to Cauty?s example (Cauty, 1994 [3]) of a metric vector space which is not an absolute retract. It is shown that Cauty?s space can be refined to the effect that the completion of the refined space can be isomorphically embedded as a subspace of an F-space which itself is an absolute retract.