We show the consistency of ZFC + ”there is no NWD-ultrafilter on ω”, which means: for every non-principal ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for every nowhere dense set A of reals, {n : f(n) ∈ A} / ∈ D. This answers a question of van Douwen, which was put in more general context by Baumgartner. ∗ The researc...

We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k-coloured Milliken-Taylor ultrafilters have at least k + 1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Mill...

In this paper, closed ideals in Cc(X), the functionally countable subalgebra of C(X), with mc-topology, is studied. We show that ifX CUC-space, then C*c(X) uniform norm-topology a Banach algebra. Closed Cc(X) as modified analogue C(X) m-topology are characterized. For zero-dimensional space X, we proper ideal if and only it an intersection maximal Cc(X). It also shown every mc-topology X P-spac...

Even a functor without an adjoint induces a monad, namely, its codensity monad; this is subject only to the existence of certain limits. We clarify the sense in which codensity monads act as substitutes for monads induced by adjunctions. We also expand on an undeservedly ignored theorem of Kennison and Gildenhuys: that the codensity monad of the inclusion of (finite sets) into (sets) is the ult...

This paper will remind the reader of neighborhood semantics for modal logics, compare them with relational semantics, and then look at some questions about neighborhood semantics that have been answered, and some which are still open. We will then introduce “ultrafilter semantics,” a way of expressing all sets over a canonical frame in an ‘effable’ way. This provides us with a conceptually easy...

We introduce the decomposability spectrum KD = {λ ≥ ω|D is λ-decomposable} of an ultrafilter D, and show that Shelah’s pcf theory influences the possible values KD can take. For example, we show that if a is a set of regular cardinals, μ ∈ pcf a, the ultrafilter D is |a|-complete and KD ⊆ a, then μ ∈ KD. As a consequence, we show that if λ is singular and for some λ < λ KD contains all regular ...