David Milovich

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Could it be that the above questions have resisted solution because we haven’t tapped into the full strength of “proper?” Let us focus on the second question for a moment. Consider the following analogy. Recall that a forcing Q satisfies the countable chain condition (c.c.c.) if every uncountable collection of conditions in Q contains two compatible(More)
We construct a path-connected homogenous compactum with cellularity 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 Ti spaces(More)
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],⊆〉. We also show that 〈U ,⊇∗〉 ≡T 〈[c] ,⊆〉 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.
Extending some results of Malykhin, we prove several independence results about base properties of βω \ ω and its powers, especially the Noetherian type Nt(βω \ ω), the least κ for which βω \ ω has a base that is κ-like with respect to containment. For example, Nt(βω \ ω) is at least s, but can consistently be that ω1, c, c, or strictly between ω1 and c.(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)
We show that, given finitely many line-segment mirrors in the plane, that do not touch, and an arbitrary point source of light, if all angles made by lines parallel to mirrors are rational multiples of π, then all but countably many emitted light beams escape. This result is shown to imply that, for a given point source of light, a randomly chosen(More)
We define branch product topologies, a new product construction. Branch product topologies generalize sum topologies and product topologies. We investigate criteria for branch products or iterations of branch products to preserve various topological properties, focusing on the separation axioms and compactness. In particular, we generalize the Tychonoff(More)
The Noetherian type of a space is the least κ for which the space has a κop-like base, i.e., a base in which no element has κ-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with(More)