Paolo Lipparini
We generalize the results from "P. Lipparini, Productive $[λ,μ]$-compactness and regular ultrafilters, Topology Proceedings, 21 (1996), 161--171"; in particular the present results apply to singular cardinals, too.
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We find many conditions equivalent to the model-theoretical property $λ\stackrelκ{\Rightarrow} μ$ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices.
We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.
We discuss the existence of complete accumulation points of sequences in products of topological spaces. Then we collect and generalize many of the results proved in Parts I, II and IV. The present Part VI is complementary to Part V to the effect that here we deal, say, with uniformity, complete accumulation points and $ κ$-$(λ)$-compactness, rather than with regularity, $[ λ, μ]$-compactness and $ κ$-$ (λ, μ)$-compactness. Of course, if we restrict ourselves to regular cardinals, Parts V (for $ λ= μ$) and Part VI essentially coincide.
We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness are equivalent. Any product of initially $λ$-compact generalized ordered topological spaces is still initially $λ$-compact. On the other hand, preservation under products of certain compactness properties are independent from the usual axioms for set theory.
High dimensional countable compactness and ultrafilters
该研究定义了高维序列的极限点与α-可数紧性概念，通过反例区分不同紧性类别，构造出满足特定性质的βω子空间解答公开问题，并引出新型超滤子。
We discuss some notions of compactness and convergence relative to a specified family F of subsets of some topological space X. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in which F is the family of all singletons of X, in which case we get back the more usual notions. (2) The case in which F is the family of all nonempty open subsets of X, in which case we get notions related to pseudocompactness. A large part of the results in this note are known in particular case (1); the results are, in general, new in case (2). As an exampl…
We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary estensions. We also have results involving products of possibly uncountable regular cardinals.
Local versions of measurability have been around for a long time. Roughly, one splits the notion of $μ$-completeness into pieces, and asks for a uniform ultrafilter over $μ$ satisfying just some piece of $μ$-completeness. Analogue local versions of weak compactness are harder to come by, since weak compactness cannot be defined by using a single ultrafilter. We deal with the problem by restricting just to a subset $P$ of all the partitions of $μ$ into $<μ$ classes and asking for some ultrafilter $D$ over $μ$ such that no partition in $P$ disproves the $μ$-completeness of $D$. By making $P$ var…
We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.
Characterizations of $\mathbb{N}$-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
本文在 ZF 集合论框架下考察 Herrlich-Chew 定理与 Hewitt 定理，证明前者无需选择公理，后者在可数多选公理下成立，并给出其在 ZF 中有效的修正形式及应用。
We introduce a covering notion depending on two cardinals, which we call $\mathcal O $-$ [ μ, λ]$-compactness, and which encompasses both pseudocompactness and many other generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to $\mathcal O $-$ [ ω, ω ]$-compactness. We provide several characterizations of $\mathcal O $-$ [ μ, λ]$-compactness, and we discuss its connection with $D$-pseudocompactness, for $D$ an ultrafilter. We analyze the behaviour of the above notions with respect to products. Finally, we show that our results hold in a more g…
Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeabili…
Contents: 1. Editor's note 2. Research announcements 2.1. Pseudocompact group topologies with no infinite compact subsets 2.2. Selective coideals on (FIN[1]k) 2.3. Entire functions mapping uncountable dense sets of reals onto each other monotonically 2.4. Effective refining of Borel coverings 2.5. Symmetry and colorings: Some results and open problems 2.6. Many partition relations below density 2.7. Lindelof indestructibility, topological games and selection principles 2.8. Locally precompact groups: (Local) realcompactness and connectedness 2.9. The group Aut is Roelcke precompact 2.10. An in…
We prove that the existence of $\mathfrak{c}$ incomparable selective ultrafilters implies the existence of a Wallace semigroup whose cube is countably compact. In addition, assuming the existence of $2^{\mathfrak c}$ incomparable selective ultrafilters and $2^{< 2^{\mathfrak{c}}} = 2^{\mathfrak{c}}$, we obtain torsion-free topological groups with respect to Comfort's question on the countable compactness of (infinite) powers of a topological group.
We use the ultrafilter-convergence axiomatics for topological spaces to motivate in detail a gentle categorical introduction, first to Barr's Set-based relational T-algebras, and then to Burroni's T-preorders internal to a category C, here called T-spaces in C, for a monad T on C that substitutes the ultrafilter monad on Set. Within these settings one finds not only the notions of compactness and Hausdorff separation, originally due to Manes, but also that of complete regularity. Based on a somewhat hidden result by Burroni, the main theorem of this paper establishes an external fibrational ch…
A generalized topology in a set $X$ is a collection $\text{Cov}_X$ of families of subsets of $X$ such that the triple $(X,\bigcup \text{Cov}_X,\text{Cov}_X)$ is a generalized topological space in the sense of Delfs and Knebusch. In this work, notions of topological and admissible compactness of generalized topologies are introduced to begin and investigate a theory of compactifications, in particular, of Wallman type in the category of weakly normal generalized topological spaces. Among other facts, we prove in ZF that the ultrafilter theorem (in abbreviation UFT) holds if and only if all Wall…
If $\mathcal P$ is a family of filters over some set $I$, a topological space $X$ is \emph{sequencewise $\mathcal P$-\brfrt compact} if, for every $I$-indexed sequence of elements of $X$, there is $F \in \mathcal P$ such that the sequence has an $F$-limit point. Countable compactness, sequential compactness, initial $κ$-compactness, $[ λ,μ]$-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise $\mathcal P$-compactness, for appropriate choices of $\mathcal P$. We show that sequencewise $\mathcal P$-compactness is preserved under taking products if and …
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