Totally imperfect Menger sets

Seminarium: 
Topologia
Osoba referująca: 
Piotr Szewczak (UKSW)
Data: 
wtorek, 6. Czerwiec 2023 - 17:15
Sala: 
A.4.1 C-19 (Politechnika Wrocławska)
Opis: 
A set of reals X is Menger if for any countable sequence of open covers of X one can pick finitely many elements from every cover in the sequence such that the chosen sets cover X. Any set of reals of cardinality smaller than the dominating number d is Menger and there is a non-Menger set of cardinality d. By the result of Bartoszyński and Tsaban, in ZFC, there is a totally imperfect (with no copy of the Cantor set inside) Menger set of cardinality d. We solve a problem, whether there is such a set of cardinality continuum. Using an iterated Sacks forcing and topological games we prove that it is consistent with ZFC that d