Covering a bounded set of functions by an increasing chain of slaloms

The Hurewicz covering property and slaloms in the Baire space

According to a result of Kocinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals $X$ satisfies the Hurewicz property if, and only if, each large open cover of $X$ contains a groupable subcover. This solves in the affirmative a problem of Scheepers. The proof uses a rigorously justified a…

SPM Bulletin 17

CONTENTS: Lecce Workshop presentations available online; Borel cardinalities below c_0; Hereditarily non-topologizable groups; A hodgepodge of sets of reals; Random gaps; Covering a bounded set of functions by an increasing chain of slaloms; Baire-one mappings contained in a usco map; Applications of k-covers II; Additivity numbers of covering properties; Combinatorial images of sets of reals and semifilter trichotomy; Another algebraic equivalent of the Continuum Hypothesis; A connection between decomposable ultrafilters and possible cofinalities II; Game Approach to Universally Kuratowski-Ul…

On the length of chains of proper subgroups covering a topological group

We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-sigma-bounded topological group G admits an increasing chain <G_a : a < b(L)> of its proper subgroups such that: (i) U_{a in b(L)} G_a=G; and $(ii)$ For every sigma-bounded subgroup H of G there exists a such that H is a subset of G_a. In case of the group Sym(w) of all permutations of w with the topology inherited from w^w this improves upon earlier results of S. Thomas.

Coverings by open cells

We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.

迹 $\mathrm{W}^*$-代数中类型的覆盖熵

Covering entropy for types in tracial $\mathrm{W}^*$-algebras

结合自由概率论与模型论技术，研究迹 $\mathrm{W}^*$-代数微观态空间的覆盖熵，证明若 $h(\mathcal{N}:\mathcal{M}) \geq 0$ 则存在保持熵近似的矩阵超积嵌入，并推断所有此类嵌入自同构等价时 $\mathcal{M}$ 强 $1$-有界。

On effective sigma-boundedness and sigma-compactness

We prove several theorems on sigma-bounded and sigma-compact pointsets. We start with a known theorem by Kechris, saying that any lightface Σ^1_1 set of the Baire space either is effectively sigma-bounded (that is, covered by a countable union of compact lightface Δ^1_1 sets), or contains a superperfect subset (and then the set is not sigma-bounded, of course). We add different generalizations of this result, in particular, 1) such that the boundedness property involved includes covering by compact sets and equivalence classes of a given finite collection of lightface Δ^1_1 equivalence relatio…