Piotr Koszmider

On decompositions of Banach spaces of continuous functions on Mrówka’s spaces

It is well known that if K is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space C(K) of continuous functions on K has complemented copies of c 0 , i.e., C(K) ∼ c 0 ○+ X ∼ c 0 ○+ c 0 ○+ X ∼ c 0 ○+ C(K). We address the question if this could be the only type of decompositions of C(K)? c 0 into infinite-dimensional summands for K infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martins axiom we construct an example of Mrowkas space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.

On strong chains of uncountable functions

For functionsf,g:ω1 → ω1, where ω1 is the first uncountable cardinal, we write thatf≪g if and only if {ξ ∈ ω1 :f(ξ)≥g(ξ)} is finite. We prove the consistency of the existence of a well-ordered increasing ≪-chain of length ω12, solving a problem of A. Hajnal. The methods previously developed by us involveforcing with side conditions in morasses which is a variation on Todorcevicsforcing with models as side conditions. The paper is self-contained and requires from the reader knowledge of Kunens textbook and some basic experience with proper forcing and elementary submodels.

Forcing minimal extensions of Boolean algebras

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk’s method more flexible, obtaining, for every cardinal λ of uncountable cofinality, a consistent example of a Boolean algebra Aλ whose every infinite homomorphic image is of cardinality λ and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight λ). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra A can be strictly less than the minimal size of a homomorphic image of A, answering a question of J. D. Monk. 2) We prove that for every cardinal of uncountable cofinality it is consistent that 2ω = λ and both Aλ and Aω1 exist. 3) By combining these algebras we obtain many examples that answer questions of J.D. Monk. 4) We prove the consistency of MA + ¬CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangel′skii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character. 5) We prove that for any cardinal λ of uncountable cofinality it is consistent that there is a countably tight Boolean algebra A with a distinguished ultrafilter ∞ such that for every a 63 ∞ the algebra A|a is countable and ∞ has hereditary character λ.

A Lindelöf space with no Lindelöf subspace of size ℵ_1

A consistent example of an uncountable Lindelof T 3 (and hence normal) space with no Lindelof subspace of size N 1 is constructed. It remains unsolved whether extra set-theoretic assumptions are necessary for the existence of such a space. However, our space has size N 2 and is a P-space, i.e., G δ s are open, and for such spaces extra set-theoretic assumptions turn out to be necessary.