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We construct an example of a Banach space E such that every weakly compact subset of E is bisequential and E contains a weakly compact subset which cannot be embedded in a Hilbert space equipped with the weak topology. This answers a question of Nyikos.

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions $c_p\left(X\right)$ onto $c_p\left(X\right)$ × ℝ$.Inparticular,$cp(X)$isnotlinearlyhomeomorphicto$cp(X)$\times ℝ$. One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight ${\omega }_{\omega }$ and with the third derived set $K^{\left(3\right)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c₀\left({\omega }_{\omega }\right)$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

We consider the class of compact spaces $K_A$ which are modifications of the well known double arrow space. The space $K_A$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_A$ spaces and on the isomorphic classification of the Banach spaces $C\left(K_A\right)$.

We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into ${\ell }_{\infty }/c₀$. We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into ${\ell }_{\infty }/c₀$, but fails to embed isometrically....

We study extension operators between spaces of continuous functions on the spaces $\sigma ₙ\left(2^X\right)$ of subsets of X of cardinality at most n. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator $T:C\left(\lambda B_H\right)\to C\left(\mu B_H\right)$.

In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{\sigma \delta \sigma }$-subset of X and contains a retract R so that $R\times E^{\omega }$ is not homeomorphic to $E^{\omega }$. This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

We prove that for each countably infinite, regular space X such that $C_p\left(X\right)$ is a $Z_{\sigma }$-space, the topology of $C_p\left(X\right)$ is determined by the class $F_0\left(C_p\left(X\right)\right)$ of spaces embeddable onto closed subsets of $C_p\left(X\right)$. We show that $C_p\left(X\right)$, 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 $F_0\left(C_p\left(X\right)\right)=M_{\alpha }$. For each ordinal α ≥ 2, we provide an example $X_{\alpha }$ such that $C_p\left(X_{\alpha }\right)$ is homeomorphic to ${\Omega }_{\alpha }$.