Daria Perkowska
Daria Perkowska: Non-meager filters
05/12/23 11:08
Tuesday, December 5, 2023 17:00
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Daria Perkowska
Title: Non-meager filters
Abstract: In the talk I will consider filters on \(\omega\) in the measurability (and complexity) context. Also, one can distinguish some natural subclasses of non-meager filters. We say that a filter \(F\) is ccc if \(P(\omega) /F\) is ccc. Similarly, we say that a filter supports a measure if there is a probability measure \(\mu\) on \(\omega\) such that \(F = \{A: \mu(A)=1\}\). I will show that every ultrafilter supports a measure, every measure supporting filter is ccc and every ccc filter is non-meager. So, one can think about these notions as forming some hierarchy of complexity of filters. This hierarchy is strict. Next I will show that for every ultrafilter from the forcing extension (by \(\mathbb{A}\)), there is a ground model filter F such that the ultrafilter extends F and there is an injective Boolean homomorphism \(\varphi: P(\omega) /F \to \mathbb{A}\).
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Daria Perkowska
Title: Non-meager filters
Abstract: In the talk I will consider filters on \(\omega\) in the measurability (and complexity) context. Also, one can distinguish some natural subclasses of non-meager filters. We say that a filter \(F\) is ccc if \(P(\omega) /F\) is ccc. Similarly, we say that a filter supports a measure if there is a probability measure \(\mu\) on \(\omega\) such that \(F = \{A: \mu(A)=1\}\). I will show that every ultrafilter supports a measure, every measure supporting filter is ccc and every ccc filter is non-meager. So, one can think about these notions as forming some hierarchy of complexity of filters. This hierarchy is strict. Next I will show that for every ultrafilter from the forcing extension (by \(\mathbb{A}\)), there is a ground model filter F such that the ultrafilter extends F and there is an injective Boolean homomorphism \(\varphi: P(\omega) /F \to \mathbb{A}\).