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Search: MSC category 54H05
( Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] )
1. CMB 2015 (vol 58 pp. 334)
 Medini, Andrea

Countable Dense Homogeneity in Powers of Zerodimensional 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¡ndezGutiÃ©rrez,
HruÅ¡Ã¡k and van Mill, using a technique of Medvedev, we
construct a nonPolish 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 zerodimensional 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)
3. CMB 2010 (vol 54 pp. 180)
 Spurný, J.; Zelený, M.

Additive Families of Low Borel Classes and Borel Measurable Selectors
An important conjecture in the theory of Borel sets in nonseparable
metric spaces is whether any pointcountable Boreladditive family in
a complete metric space has a $\sigma$discrete refinement. We confirm the conjecture for
pointcountable $\mathbf\Pi_3^0$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, Boreladditive family, measurable selection Categories:54H05, 54E35 
