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Universal sets for ideals

Cieślak A., Michalski M.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 2

157--166

We consider the notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classical ideals like the null subsets of 2ω and the meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for E, the σ-ideal generated by closed null subsets of 2ω, and for some ideals connected with forcing notions: the Kσ subsets of ωω and the Laver ideal. We also consider Fubini products of ideals and show that there are Σ03 universal sets for N[symbol]M and M[symbol]N.

Historical remarks on equivalence of functions

Lipecki Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

Vol. 65, no. 1

93--95

A theorem of E. Szpilrajn (Marczewski) (1936) on equivalent functions is recalled and its relation to subsequent results published by M. Morayne and C. Ryll-Nardzewski (1999) and M. Kysiak (2005) is briefly discussed.

A Polish AR-Space with no nontrivial isotopy

Dobrowolski T.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 1

67-74

The Polish space Y constructed in [vM1] admits no nontrivial isotopy. Yet, there exists a Polish group that acts transitively on Y.

On some theorem on measurable selection

Shragin I.V.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2007

Vol. 47, [Z] 2

221-225

A new proof of some results of V.L. Levin on measurability of projection and on measurable selection is given.

On countable dense and strong local homogeneity

Mill J. van

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 4

401-408

We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.