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1. CMB Online first
Countable dense homogeneity in powers of zero-dimensional definable spaces We show that, for a coanalytic subspace $X$ of $2^\omega$, the
countable dense homogeneity of $X^\omega$ is equivalent to $X$
being Polish. This strengthens a result of HruÅ¡Ã¡k and Zamora
AvilÃ©s. Then, inspired by results of HernÃ¡ndez-GutiÃ©rrez,
HruÅ¡Ã¡k and van Mill, using a technique of Medvedev, we
construct a non-Polish subspace $X$ of $2^\omega$ such that $X^\omega$
is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer
to a question of HruÅ¡Ã¡k and Zamora AvilÃ©s. Furthermore,
since our example is consistently analytic, the equivalence result
mentioned above is sharp. Our results also answer a question
of Medini and Milovich. Finally, we show that if every countable
subset of a zero-dimensional separable metrizable space $X$ is
included in a Polish subspace of $X$ then $X^\omega$ is countable
dense homogeneous.
Keywords:countable dense homogeneous, infinite power, coanalytic, Polish, $\lambda'$-set Categories:54H05, 54G20, 54E52 |
2. CMB 2011 (vol 54 pp. 302)
Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces Let $X$ be a separable non-reflexive Banach space. We show that there
is no Borel class which contains the set of norm-attaining functionals
for every strictly convex renorming of $X$.
Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class Categories:46B20, 54H05, 46B10 |
3. CMB 2010 (vol 54 pp. 180)
Additive Families of Low Borel Classes and Borel Measurable Selectors
An important conjecture in the theory of Borel sets in non-separable
metric spaces is whether any point-countable Borel-additive family in
a complete metric space has a $\sigma$-discrete refinement. We confirm the conjecture for
point-countable $\mathbf\Pi_3^0$-additive families, thus generalizing results of
R. W. Hansell and the first author. We apply this result to the
existence of Borel measurable selectors for multivalued mappings of
low Borel complexity, thus answering in the affirmative a particular
version of a question of J. Kaniewski and R. Pol.
Keywords:$\sigma$-discrete refinement, Borel-additive family, measurable selection Categories:54H05, 54E35 |