An ultrafilter p on ω is said to be discrete if, given any function f:ω→X completely regular Hausdorff space, there an A∈p such that f(A) discrete. Basic properties of ultrafilters are studied. Three intermediate classes spaces R1⊂R2⊂R3 between the class F-spaces and van Douwen's βω-spaces introduced. It proved no product infinite compact R2-spaces homogeneous; moreover, under assumption d=c, h...

Dead-end ultrafiltration (DEUF) is an alternative approach to tangential-flow hollow-fiber ultrafiltration that can be readily employed under field conditions to recover microbes from water. The hydraulics of DEUF and microbe recovery for a new DEUF method were investigated using 100-liter tap water samples. Pressure, flow rate, and temperature were investigated using four hollow-fiber ultrafil...

Let X be a metric space, {xi}i∈N a sequence of points in X (thought of as a sequence of centers), and {di}i∈N a sequence of scaling factors such that di → ∞, and an ultrafilter ω. I’m not going to get into the details of ultrafilters here, but the main property of an ultrafilter is that it lets you define the ultralimit limω ai of any bounded sequence ai. Ultralimits use the axiom of choice to ...

u1 = {Z | there are finitely many sets X1, . . . Xk such that Z = X1 ∩ · · · ∩Xk}. That is, u1 is the set of finite intersections of sets from u0. Note that u0 ⊆ u1, since u0 has the finite intersection property, we have ∅ 6∈ u1, and by definition u1 is closed under finite intersections. Now, define u2 as follows: u′ = {Y | there is a Z ∈ u1 such that Z ⊆ Y } ∗UMD, Philosophy. Webpage: ai.stanf...

Let PR(X) denote the hyperspace of nonempty finite subsets a topological spaceX with Pixley-Roy topology. In this paper, by introducing closed-miss-finite networks and using principle ultrafilters, we proved that following statements are equivalent for space X: (1) is weakly Rothberger; (2) X satisfies S1(?rcf ,?wrcf ); (3) separable ? {x} S1(?cf ,?wcf ) each x X; (4) principal ultrafilter F [x...

A locally compact T2-space is called a Franklin-Rajagopalan space (or FR-space) provided it has a countable discrete dense subset whose complement is homeomorphic to an ordinal with the order topology. We show that (1) every sequentially compact FR-space X can be identified with a space constructed from a tower T on w (X = X(T)), and (2) for an ultrafilter u on w, a sequentially compact FR-spac...

We show that while the length ω iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Př́ıkrý forcing, it is consistent that no iteration of length greater than ω (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a general...

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened ...

Let T1, T2 be countable first-order theories, Mi |= Ti, and D any regular ultrafilter on λ ≥ א0. A longstanding open problem of Keisler asks when T2 is more complex than T1, as measured by the fact that for any such λ,D, if the ultrapower (M2)/D realizes all types over sets of size ≤ λ, then so must the ultrapower (M1)/D. In this paper, building on the author’s prior work [11] [12] [13], we sho...