Witold Marciszewski

On topological classification of function spaces _{}() of low Borel complexity

We prove that if X is a countable nondiscrete completely regular space such that the function space Cp(X) is an absolute Fa-set, then Cp(X) is homeomorphic to v°°, where a = {(xi) E R°°:xi = O for all but finitely many i} . AS an application we answer in the negative some problems of A. V. Arhangelskil by giving examples of countable completely regular spaces X and Y such that X fails to be a bR-space and a k-space (and hence X is not a kc,,-space and not a sequential space) and Y fails to be an 80-space while the function spaces Cp(X) and Cp(Y) are homeomorphic to Cp(X) for the compact metric space 3S = {0} U {n1: n = 1, 2, . . . } .

ABSOLUTE BOREL SETS AND FUNCTION SPACES

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolik for characterizing Cechcomplete spaces. We also show that the absolute Borel class of X is determined by the uniform structure of the space of continuous functions Cp (X); however the case of absolute G6 metric spaces is still open. More precisely, we prove that, for metrizable spaces X and Y, if 1, then Y is also an absolute Borel set of the same class. This result is new even if P is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the Cech-completeness of a metric space X is determined by the linear structure of Cp(X).

On some problems concerning Borel structures in function spaces

It is an open problem if any separable compact space K whose function space C(K) with the cylindrical σ-algebra is a standard measurable space, embeds in the space of the first Baire class functions on the Cantor set, with the pointwise topology. We prove that this is true for separable linearly ordered compacta.Other problems discussed in this note concern the Borel structures in C(K) generated by the norm, weak or pointwise topology in C(K). We give an example of a compact space K such that the weak and the pointwise topology generate different Borel structures in C(K).ResumenEs un problema abierto saber si cada espacio compacto separable K cuyo espacio de funciones C(K) con la σ-álgebra cilíndrica es un espacio medible estándar, se sumerge en el espacio de las funciones de la primera clase de Baire definidas en el conjunto de Cantor, con la topología de convergencia puntual. Aquí probamos que lo anterior es cierto para compactos separables linealmente ordenados. Discutimos en esta nota otros problemas relativos a las estructuras de Borel en C(K) generadas por la norma, topología débil o topología de convergencia puntual en C(K). Damos un ejemplo de un espacio compacto K tal que la topología débil y la topología de convergencia puntual generan estructuras de Borel diferentes en C(K).

A countable X having a closed subspace A with Cp(A) not a factor of Cp(X)

Abstract Let A be a countable space such that the function space C p ( A ) is analytic. We prove that there exists a countable space X such that X contains A as a closed subset and the function space C p ( X ) is an absolute F σδ -set. Therefore, if C p ( A ) is analytic non-Borel then C p ( A ) is not a factor of C p ( X ) and there is no continuous (or even Borel-measurable) extender e : C p ( A ) → C p ( X ) (i.e., a map such that e(f)¦A = f , for f ϵ C p ( A )). This answers a question of Arkhangelskiĭ. We also construct a countable space X such that the function space C p ( X ) is an absolute F σδ -set and X contains closed subsets A with C p ( A ) of arbitrarily high Borel complexity (or even analytic non-Borel).

On properties of metrizable spaces X preserved by t-equivalence

For a completely regular space X, denote by Cp(X) the space of continuous real valued functions on X, endowed with the pointwise convergence topology. The spaces X and Y are t-equivalent if Cp(X) and Cp(Y) are homeomorphic. It is proved that, for metrizable spaces X, the countable dimensionality is preserved by t-equivalence. It is also shown that this relation preserves absolute Borel classes greater than 2 and all projective classes.

Recent classification and characterization results in geometric topology

We announce a complete topological classification of the function spaces C (X) of Borel class not higher than 2, provided that I is a countable space. We also present a topological classification of the /c-dimensional universal pseudoboundaries and pseudointeriors in R , and we investigate under what conditions strong negligibility of crZ-sets characterizes Hilbert space manifolds.

Networks for the weak topology of Banach and Fréchet spaces

Abstract We start the systematic study of Frechet spaces which are ℵ-spaces in the weak topology. A topological space X is an ℵ 0 -space or an ℵ-space if X has a countable k-network or a σ-locally finite k-network, respectively. We are motivated by the following result of Corson (1966): If the space C c ( X ) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology is a Banach space, then C c ( X ) endowed with the weak topology is an ℵ 0 -space if and only if X is countable. We extend Corsons result as follows: If the space E : = C c ( X ) is a Frechet lcs, then E endowed with its weak topology σ ( E , E ′ ) is an ℵ-space if and only if ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Frechet lcs to be an ℵ-space in the weak topology. We prove that a reflexive Frechet lcs E in the weak topology σ ( E , E ′ ) is an ℵ-space if and only if ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if E is separable. We show however that the nonseparable Banach space l 1 ( R ) with the weak topology is an ℵ-space.

On analytic and coanalytic function spaces Cp(X)

Abstract Under the assumption ( V = L ) we construct countable completely regular spaces X and Y such that the spaces C p ( X ) and C p ( Y ) of real-valued continuous functions on X and Y , equipped with the pointwise convergence topology, are analytic noncoanalytic and they are not homeomorphic. We also give analogous examples of coanalytic nonanalytic function spaces.

On Corson compacta and embeddings of

We investigate properties of those compact spaces K for which the Banach space C(K) can be isomorphically embedded into a space C(L), where L is Corson compact. We show that in such a case K must be Corson compact provided K has some additional measure—theoretic property. The result is applicable to Rosenthal compacta and several other classes of compact spaces K.

C(K)

Abstract A Corson compactum is a compact subspace of the Σ-product of the real line: Σ (Г) ={{x ∈ R Г : supp(x) = {γ ∈ Г: x(γ) ≠ 0} is countable}. For every zero-dimensional Corson compactum K we define the order type of K to be the least ordinal α such that there are an ordinal η and an embedding h : K → Σ(η) ∩ {0, 1}η such that order type of supp(h(k)) ⩽ α for all k ∈ K. We show that zero-dimensional Eberlein compacta have order type ω and give an example of a zero-dimensional Talagrand compactum which has order type ω1, answering questions of Argyros, Mercourakis and Negrepontis. We solve a problem of Comfort and Negrepontis by showing that it is consistent with ZFC that 2ω ⩾ ω1 and there is a ccc Corson compact space without precalibre (2ω, ω1). Also we discuss the equality spread = weight for Corson compacta.