In every infinite-dimensional Frechet space X, we construct a linear sub-space E such that E is an F-sigmadeltasigma-subset of X and contains a retract R so that R x E-omega is not homeomorphic to E-omega. This shows that Torunczyk's Factor Theorem fails in the Borel case.

We are concerned with C-A(X), the space of continuous real valued functions on X considered with the topology of pointwise convergence on A, where A is a countable dense subset of X. We focus on the Borel and the topological classifications of the spaces CA(X). For example, we prove that for countable nondiscrete X, C-A(X) is homeomorphic to sigma(omega), the countable product of sigma = {(x(i)) epsilon R(omega) \ x(i) = 0 a.e.}, provided C-A(X) epsilon F-sigma delta.

We determine the topological structure of the subsets of the hyperspace 2(R2) consisting of absolute retracts and of absolute neighborhood retracts respectively.

The title statement is proved. Similar results for arbitrary Banach spaces are obtained in both the real-analytic and the Cp settings.

A notion of true dimension theory is defined to which is assigned a dimension function D. We consider those D which have an enhanced Bockstein basis; these include D = dim and D = dim(G), for any abelian group G. We prove that for each countable polyhedron K, the set of compacta X is-an-element-of 2Q with K is-an-element-of AE({X}) is a G(delta)-subspace. We apply this fact to show that the hyperspace of the Hilbert cube Q consisting of compacta (or continua) X with D(X) less-than-or-equal-to n is a G(delta)-subspace. Let D(greater-than-or-equal-to n) (resp., D(greater-than-or-equal-to n) and C(Q)) denote the space of compacta X (resp., continua) with D(X) greater-than-or-equal-to n. We prove that {D(greater-than-or-equal-to n)n=1 infinity and {D(greater-than-or-equal-to n) and C(Q)}n=2 infinity are absorbing sequences for sigma-compact spaces. This yields that each D(greater-than-or-equal-to n) and D(greater-than-or-equal-to n+1) and C(Q) (n greater-than-or-equal-to 1) is homeomorphic to the pseudoboundary B of Q; their respective complements are homeomorphic to the pseudointerior of Q; and the intersections and n D(greater-than-or-equal-to n), and n D(greater-than-or-equal-to n) and C(Q) are homeomorphic to B(infinity), the absorbing set for the class of F(sigmadelta)-sets. Results for the hyperspaces of compacta X for which D(X) greater-than-or-equal-to n uniformly are also obtained.

Using the l 2 {l^2} -products we find pre-Hilbert spaces that are absorbing sets for all Borelian classes of order α ≥ 1 \alpha \geq 1 . We also show that the following spaces are homeomorphic to Σ ∞ \Sigma ^\infty , the countable product of the space Σ = { ( x n ) ∈ R ∞ : ( x n ) \Sigma = \{(x_n) \in R^\infty : (x_n) is bounded}: (1) every coordinate product ∏ C H n \prod _C H_n of normed spaces H n H_n in the sense of a Banach space C C , where each H n H_n is an absolute F σ δ F_{\sigma \delta } -set and infinitely many of the H n H_n ’s are Z σ {Z_\sigma } -spaces, (2) every function space L ~ p = ∩ p ′ > p L p ′ \tilde {L}^p = \cap _{p\prime >p}L^{p\prime } with the L q {L^q} -topology, 0 > q > p ≤ ∞ 0>q>p \leq \infty , (3) every sequence space l ~ p = ∩ p > p ′ l p ′ {\tilde l^p} = { \cap _{p > p\prime }}{l^{p\prime }} with the l q l^q -topology, 0 ≤ p > q > ∞ 0 \leq p > q > \infty . We also note that each additive and multiplicative Borelian class of order α ≥ 2 \alpha \geq 2 , each projective class, and the class of nonprojective spaces contain uncountably many topologically different pre-Hilbert spaces which are Z σ Z_\sigma -spaces.

We prove that for each countably infinite, regular space X such that C(p)(X) is a Z(sigma)-space, the topology of C(p)(X) is determined by the class F0(C(p)(X)) of spaces embeddable onto closed subsets of C(p)(X). We show that C(p)(X), whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set OMEGA(alpha) for the multiplicative Borel class M(alpha) if F0(C(p)(X)) = M(alpha). For each ordinal alpha greater-than-or-equal-to 2, we provide an example X(alpha) such that C(p)(X(alpha)) is homeomorphic to OMEGA(alpha).

We prove that if X is a countable nondiscrete completely regular space such that the function space C (X) is an absolute FaS-set, then C (X) is homeomorphic to <7°° , where a = {(x¡) e R00:*, = 0 for all but finitely many i} .As an application we answer in the negative some problems of A. V. Arhangel'skii by giving examples of countable completely regular spaces X and y such that X fails to be a 6R-space and a fc-space (and hence X is not a km-space and not a sequential space) and Y fails to be an N0-space while the function spaces C (X) and CJY) are homeomorphic to CJX) for the compact metric space X = {0} U {n~ : n = 1, 2, ... } .

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. Arhangel'skil 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, . . . } .