Wiesław Kubiś

Descriptive topology in selected topics of functional analysis

Preface.- 1. Overview.- 2. Elementary Facts about Baire and Baire-Type Spaces.- 3. K-analytic and quasi-Suslin Spaces.- 4. Web-Compact Spaces and Angelic Theorems.- 5. Strongly Web-Compact Spaces and a Closed Graph Theorem.- 6. Weakly Analytic Spaces.- 7. K-analytic Baire Spaces.- 8. A Three-Space Property for Analytic Spaces.- 9. K-analytic and Analytic Spaces Cp(X).- 10. Precompact sets in (LM)-Spaces and Dual Metric Spaces.- 11. Metrizability of Compact Sets in the Class G.- 12. Weakly Realcompact Locally Convex Spaces.- 13. Corsons Propery (C) and tightness.- 14. Frechet-Urysohn Spaces and Groups.- 15. Sequential Properties in the Class G.- 16. Tightness and Distinguished Frechet Spaces.- 17. Banach Spaces with Many Projections.- 18. Spaces of Continuous Functions Over Compact Lines.- 19. Compact Spaces Generated by Retractions.- 20. Complementably Universival Banach Space.- Index.

A proof of uniqueness of the Gurariĭ space

We present a short and elementary proof of isometric uniqueness of the Gurariĭ space.

Fraïssé sequences: category-theoretic approach to universal homogeneous structures

We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.

Small Valdivia compact spaces

Abstract We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight ⩽ ℵ 1 is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight ⩽ ℵ 1 is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.

Tightness and Distinguished Fréchet Spaces

In this chapter, we apply the concept of tightness to study distinguished Frechet spaces. We show that a Frechet space is distinguished if and only if its strong dual has countable tightness. This approach to studying distinguished Frechet spaces leads to a rich supply of (DF)-spaces whose weak ∗ duals are quasi-Suslin but not K-analytic. The small cardinals \(\mathfrak{b}\) and \(\mathfrak{d}\) will be used to improve the analysis of Kothe’s echelon nondistinguished Frechet space λ 1(A).

Convex decompositions in the plane and continuous pair colorings of the irrationals

AbstractWe address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ2 which is not presentable as a countable union of convex sets satisfies the following dichotomy:(1)There is a perfect nonemptyP⊆S so that |C∩P|<3 for every convexC⊆S. In this case coveringS by convex subsets ofS is equivalent to coveringP by finite subsets, hence no nontrivial convex covers ofS can exist.(2)There exists a continuous pair coloringf: [N]2→{0, 1} of the spaceN of irrational numbers so that coveringS by convex subsets is equivalent to coveringN byf-monochromatic sets. In this case it is consistent thatS has a convex cover of cardinality strictly smaller than the continuumc in some forcing extension of the universe. We also show that iff: [N]2→{0, 1} is a continuous coloring of pairs, and no open subset ofN isf-monochromatic, then the least numberκ off-monochromatic sets required to coverN satisfiesK+>-c. Consequently, a closed subset of ℝ2 that cannot be covered by countably many convex subsets, cannot be covered by any number of convex subsets other than the continuum or the immediate predecessor of the continuum. The analogous fact is false for closed subsets of ℝ3.

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

Abstract We construct a compact linearly ordered space K ω 1 of weight ℵ 1 , such that the space C ( K ω 1 ) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, K ω 1 is a continuous image of a Valdivia compact and every separable subspace of C ( K ω 1 ) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.

Complementation in spaces of continuous functions on compact lines

Abstract We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton.

On some classes of Lindelöf Σ-spaces

Abstract We consider special subclasses of the class of Lindelof Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class L Σ ( ⩽ κ ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ ⩽ ω . In the case κ = ω , the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tkacenko. The case κ = 1 corresponds to the spaces of countable network weight, but even the case κ = 2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ 1 is in the class L Σ ( ⩽ ω ) , answering a question of Tkachuk. As well, we study whether certain compact spaces in L Σ ( ⩽ ω ) have dense metrizable subspaces, partially answering a question of Tkacenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.