Free Boolean algebras and nowhere dense ultrafilters

Abstract

An analogue of Mathias forcing is studied in connection of free Boolean algebras and nowhere dense ultraÿlters. Some applications to rigid Boolean algebras are given.

Strictly positive measures on Boolean algebras

“…We shall refer to this space as the Simon space. (UDSP spaces were previously known to exist under various set-theoretic axioms, see [4] and [18]). For completeness, at the end of this section we give a somewhat simplified ZF C construction of such a space, still based on Simon's ideas.…”

“…For instance, if F is any nonprincipial P -filter in P (ω) then such a topology τ (F) can be defined by declaring that a set U ⊆ ω <ω open if for every s ∈ U the set {n : t n ∈ U } is in F. It was noticed by Boban Veličković that one can prove that the Stone-Čech compactification of τ (F) gives a space with U DSP by an argument analogous to the one presented above. For this one can use a description of clopen sets in τ (F), see B laszczyk & Szymański [4]. It follows that if F is a P -point ultrafilter then we obtain a U DSP space which is in addition extremally disconected (Corollary 13 in [4]).…”

“…For this one can use a description of clopen sets in τ (F), see B laszczyk & Szymański [4]. It follows that if F is a P -point ultrafilter then we obtain a U DSP space which is in addition extremally disconected (Corollary 13 in [4]). …”

Pλ-sets and skeletal mappings

“…For every indexed collection F = (F t : t ∈ Seq) of filters we define the F-topology on Seq as follows: The F-topologies on Seq were introduced by Szymański [12] and Trnková [13] and studied e.g. in [2], [5], [7], [8], [14]. A review of F-topologies and their generalizations can be found in [1].…”

How high can Baumgartner’s $${\mathcal{I}}$$ I -ultrafilters lie in the P-hierarchy?

“…Let us mention some of the most important papers in this subject from our point of view: Błaszczyk [2], Brendle [3], Laflamme [18], Shelah [19,20]. The theory of I-ultrafilters on ω was developed by Flašková [8][9][10][11] in a series of articles, as well as in her Ph.D. thesis [12].…”