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In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2 ω = ω 2 and that L(P(ω 1)) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that (κ) fails for all… (More)

- DAVID MILOVICH
- 2007

We construct a path-connected homogenous compactum with cel-lularity c that is not homeomorphic to any product of dyadic compacta and first countable compacta. We also prove some closure properties for classes of spaces defined by various connectifiability conditions. One application is that every infinite product of infinite topological sums of T i spaces… (More)

- DAVID MILOVICH
- 2008

Motivated by a question of Isbell, we show that ♦ implies there is a non-P-point U ∈ βω \ ω such that neither U , ⊇ nor U , ⊇ * is Tukey equivalent to [c] <ω , ⊆. We also show that U , ⊇ * ≡ T [c] <κ , ⊆ for some U ∈ βω \ ω, assuming cf(κ) = κ ≤ p = c. We also prove two negative ZFC results about the possible Tukey classes of ultrafilters on ω. 1. Tukey… (More)

Remark. The notes which follow reflect the content of a two day tutorial which took place at the Fields Institute on 5/29 and 5/30 in 2009. Most of the content has existed in the literature for some time (primarily in the original edition of [10]) but has proved difficult to read and digest for various reasons. The only new material contained in these… (More)

- David Milovich
- 2008

Extending some results of Malykhin, we prove several independence results about base properties of βω \ ω and its powers, especially the Noetherian type N t(βω \ ω), the least κ for which βω \ ω has a base that is κ-like with respect to containment. For example, N t(βω \ ω) is at least s, but can consistently be that ω 1 , c, c + , or strictly between ω 1… (More)

- DAVID MILOVICH
- 2007

The Noetherian type of a space is the least κ such that it has a base that is κ-like with respect to containment. Just as all known homogeneous compacta have cellularity at most c, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example,… (More)

We study some cardinal invariants of an order-theoretic fashion on products and box products of topological spaces. In particular, we concentrate on the Noetherian type (N t), defined by Peregudov in the 1990s. Some highlights of our results include: (1) There are spaces X and Y such that N t(X × Y) < min{N t(X), N t(Y)}. (2) In several classes of compact… (More)

- David Milovich
- 2007