Tadeusz Dobrowolski

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, . . . } .

Applying coordinate products to the topological identification of normed spaces

Using the l 2 -products we find pre-Hilbert spaces that are absorbing sets for all Borelian classes of order α ≥ 1. We also show that the following spaces are homeomorphic to Σ∞, the countable product of the space Σ = {(x n ) ∈ R∞: (x n ) is bounded}: (1) every coordinate product Π C H n of normed spaces H n in the sense of a Banach space C, where each H n is an absolute F σδ -set and infinitely many of the H n s are Z σ -spaces, (2) every function space L p = ∩ p<P L p with the L q -topology, 0 < q < p ≤ ∞, (3) every sequence space l p = ∩ p<P l p with the l q -topology, 0 ≤ p < q < ∞. We also note that each additive and multiplicative Borelian class of order α ≥ 2, each projective class, and the class of nonprojective spaces contain uncountably many topologically different pre-Hilbert spaces which are Z σ -spaces

k-11 – Infinite-Dimensional Topology

Publisher Summary nIt is not an easy task to define the scope of infinite-dimensional (i-d) topology. Initially, the objects of the theory were i-d metric linear spaces (that is, topological vector spaces whose topology is metrizable) and their convex subsets. The standard examples of such objects are the separable i-d Hilbert space and the Hilbert cube. Beginning in the late sixties, the collection of rather isolated results concerning the topology of those spaces evolved into the theory of manifolds modeled on i-d metric linear spaces and corresponding i-d convex subsets (that is, spaces that are locally homeomorphic with the model). This approach led to the topological identification of certain i-d spaces such as topological groups, function spaces, or even spaces without any natural algebraic or convex structures like hyperspaces of compact sets. One of the most important and natural questions of i-d topology is the problem of the topological classification of i-d metric linear spaces and i-d convex sets. Specifically, the triangulation theorem states that for every Q-manifold M there exists a locally compact metrizable polyhedron K such that M and K ×Q are homeomorphic.