### A Boolean view of sequential compactness

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We give a classical proof of the theorem stating that the $\sigma $-ideal of meager sets is the unique $\sigma $-ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.

We consider the Borel structures on ordinals generated by their order topologies and provide a complete classification of all ordinals up to Borel isomorphism in ZFC. We also consider the same classification problem in the context of AD and give a partial answer for ordinals ≤ω₂.

When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications...

Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then ${f}^{-1}\left(y\right)$ is a ${K}_{\sigma}$ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of...

We prove there is a countable dense homogeneous subspace of ℝ of size ℵ₁. The proof involves an absoluteness argument using an extension of the ${L}_{\omega \u2081\omega}\left(Q\right)$ logic obtained by adding predicates for Borel sets.

A class of Banach spaces, countably determined in their weak topology (hence, WCD spaces) is defined and studied; we call them strongly weakly countably determined (SWCD) Banach spaces. The main results are the following: (i) A separable Banach space not containing ℓ¹(ℕ) is SWCD if and only if it has separable dual; thus in particular, not every separable Banach space is SWCD. (ii) If K is a compact space, then the space C(K) is SWCD if and only if K is countable.

In this paper we define and investigate a new subclass of those Banach spaces which are $K$-analytic in their weak topology; we call them strongly weakly $K$-analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weakly $K$-analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related. We show that: (i) not...

Solecki has shown that a broad natural class of ${G}_{\delta}$ ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed subset in this representation can be taken to be closed upwards.

Using Tsirelson’s well-known example of a Banach space which does not contain a copy of ${c}_{0}$ or ${l}_{p}$, for p ≥ 1, we construct a simple Borel ideal ${I}_{T}$ such that the Borel cardinalities of the quotient spaces $P\left(\mathbb{N}\right)/{I}_{T}$ and $P\left(\mathbb{N}\right)/{I}_{0}$ are incomparable, where ${I}_{0}$ is the summable ideal of all sets A ⊆ ℕ such that ${\sum}_{n\in A}1/(n+1)<\infty $. This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.