Free Ultrafilter Research Articles

New examples of MAD families with pseudocompact hyperspaces

We show that both ap=c and ⋄(b) imply the existence of MAD families with pseudocompact Vietoris hyperspace, substantially expanding the list of models where their existence is known. We also discuss some properties on the structure of fin-intersecting MAD families by providing some new examples of non-fin-intersecting MAD families with special properties. We show that the Baire number of ω⁎ is strictly larger than c if and only if for every MAD family A, Ψ(A) and its Vietoris hyperspace are p-pseudocompact for some free ultrafilter p.

Extremal structure in ultrapowers of Banach spaces

Given a bounded convex subset C of a Banach space X and a free ultrafilter {mathcal {U}}, we study which points (x_i)_{mathcal {U}} are extreme points of the ultrapower C_{mathcal {U}} in X_{mathcal {U}}. In general, we obtain that when {x_i} is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then (x_i)_{mathcal {U}} is an extreme point (respectively denting point, strongly exposed point) of C_mathcal U. We also show that every extreme point of C_{{mathcal {U}}} is strongly extreme, and that every point exposed by a functional in (X^*)_{{mathcal {U}}} is strongly exposed, provided that mathcal U is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of C_{mathcal {U}} in the case that C is a super weakly compact or uniformly convex set.

Interval Chains and Completeness in Ultrapowers of Ordered Sets

The ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.

On countable determination of the Kuratowski measure of noncompactness

A long-standing question in the theory of measures of noncompactness is that for the Kuratowski measure of noncompactness α defined on a metric space M, and for every bounded subset B⊂M, is there a countable subset B0⊂B such that α(B0)=α(B)? In this paper, we give an affirmative answer to the question above. It is done by showing that for each nonempty set B of a Banach space, there is a countable subset B0⊂B so that B is strongly finitely representable in B0, and that there is a free ultrafilter U so that B is affinely isometric to a subset of the ultrapower [co(B0)]U of co(B0).

General methods of convergence and summability

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of mathbb{N} and study the space of convergence associated with the filter. We notice that c(X) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then ell _{infty }(X) is a space of convergence associated with any free ultrafilter of mathbb{N} ; and that if X is not complete, then ell _{infty }(X) is never the space of convergence associated with any free filter of mathbb{N} . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that ell _{infty }(X) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then c(X) is a space of convergence through a certain class of such operators; and that if X is not complete, then c(X) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set mathcal{HB}(lim ):= {Tin mathcal{B} (ell _{infty }(X),X): T|_{c(X)}= lim text{ and }|T|=1} and prove that mathcal{HB}(lim ) is a face of mathsf{B} _{mathcal{L}_{X}^{0}} if X has the Bade property, where mathcal{L}_{X}^{0}:= { Tin mathcal{B} (ell _{infty }(X),X): c_{0}(X) subseteq ker (T) } . Finally, we study the multipliers associated with series for the above methods of convergence.

The coreflective hull of $$\mathcal {U}$$-Fréchet spaces

It is known that the category of all sequential spaces is a coreflective subcategory of the category of all topological spaces, and the coreflective hull of the category of all Frechet spaces is the category of all sequential spaces. In this paper, we generalize these results to $$\mathcal {U}$$ -sequential spaces and $$\mathcal {U}$$ -Frechet spaces where $$\mathcal {U}$$ is a free ultrafilter on $$\mathbb {N}$$ . Also we prove that the category of $$\mathcal {U}$$ -sequential spaces is a bi-coreflective subcategory of the category of all topological spaces where $$\mathcal {U}$$ is a free filter. As a consequence of these results, we get that some bi-coreflective subcategories of Top can be constructed by using free filters or free ultrafilters on $$\mathbb {N}$$ .

Classic Problems III

This is a survey of four problems that are “classics” in many different senses of the word, and of several related problems associated with each one. The numbering of the Classic Problems picks up where that of a similar article left off about four decades ago:IX.Is every point ofω⁎a butterfly point?X.Is there a nonmetrizable perfectly normal, locally connected continuum?XI.Is there a normal space with a σ-disjoint base that is not paracompact?XII.Is there a regular symmetrizable space with a non-Gδpoint?Several related problems are given for each classic problem. Consistency results are summarized, and there is a discussion of each problem that explains various implications among the related problems and justifies calling certain problems equivalent. For each classic problem, an appendix goes deeper into some implications and/or includes reminisces.There is a purely set-theoretic problem related to Classic Problem IX. Call a filter on a set D nowhere maximal if it does not trace an ultrafilter on any subset of D.Related Problem D. Is every free ultrafilter the join of two nowhere maximal filters?It is shown that the special case of ultrafilters on ω is actually equivalent to Classic Problem IX.

Special ultrafilters and cofinal subsets of $$({}^\omega \omega , <^*)$$

The interplay between ultrafilters and unbounded subsets of $${}^\omega \omega $$ with the order $$<^*$$ of strict eventual domination is studied. Among the tools are special kinds of non-principal (“free”) ultrafilters on $$\omega $$ . These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order $$\subset ^*$$ of almost inclusion. It is shown that the cofinality of such a base must be either $$\mathfrak {b}$$ , the least cardinality of $$<^*$$ -unbounded (“undominated”) set, or $$\mathfrak {d}$$ , the least cardinality of a $$<^*$$ -cofinal (“dominating”) set. The small uncountable cardinal $$\pi \mathfrak {p}$$ is introduced. Consequences of $$\mathfrak {b}< \pi \mathfrak {p}$$ and of $$\mathfrak r < \mathfrak {d}$$ are explored; in particular, both imply $$\mathfrak {b}< \mathfrak {d}$$ . Here $$\mathfrak r$$ is the reaping number, and is also the least cardinality of a $$\pi $$ -base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities (in other words, if $$\mathfrak {b}< \mathfrak {d}$$ and there exist simple $$P_\mathfrak {b}$$ -points and $$P_\mathfrak {d}$$ -points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on $$\omega $$ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold (or are known to fail). The strongest of these, Axiom 1, is that for every free ultrafilter $$\mathcal U$$ and for every $$<^*$$ -unbounded $$<^*$$ -chain C of increasing functions in $${}^\omega \omega $$ , C is also unbounded in the ultraproduct $${}^\omega \omega , <_\mathcal U$$ . The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem.

Weak normality properties and countable paracompactness

According to P. Nyikos, a topological space is called paranormal, if every countable discrete system of closed sets {Dn:n=1,2,3,...} may be expanded to a locally finite system of open sets {Un:n=1,2,3,...}, i.e. Dn⊂Un for each n and Dm∩Un≠∅ if and only if Dm=Dn. Using the notion of paranormality and δ-normality, we obtain some characteristics of countably paracompact spaces in classes of strongly P-sequential spaces and weakly P-sequential spaces, where P is a nonempty set of free ultrafilters.

Ultrapowers of topological spaces

We use ultrapowers to build p-compactifications (for a free ultrafilter p on ω) of topological spaces. In particular we show that the universal p-compactification derived from the Stone-Čech compactification, the one-point compactification of a locally compact space and every p-compactification of a discrete space are special cases of this construction. We will also study when the ultrapower of a topological group is a topological group.

Tangle and Ultrafilter: Game Theoretical Interpretation

This paper extends the concept of filter on X into (X, f), where X is a finite underlying set and f is a symmetric submodular function from $$2^X$$ to N. Then, we show a cryptomorphism between a free ultrafilter and co-tangle on (X, f). The paper also provides game-theoretical interpretations of a branch decomposition tree and a free ultrafilter on (X, f).

A note on bounded sets and C-compact sets in paratopological groups

In this paper, the class of weakly ω-admissible paratopological groups is introduced. We mainly show that: (1) Every bounded set in a weakly ω-admissible paratopological group is p-bounded for each free ultrafilter p on ω; (2) every C-compact set in a weakly ω-admissible paratopological group is strongly r-pseudocompact. The class of weakly ω-admissible paratopological groups contains all totally ω-narrow, precompact, locally feebly compact, weakly Lindelöf and saturated, and ω-admissible paratopological groups. It is worth mentioning that all the aforementioned spaces are not to satisfy any separation axiom. Some Sanchis, Sánchez and Tkachenko's results are improved.

On ultracompact spaces in ZF

We work in set theory without the Axiom of Choice (AC) and establish the following results:1.“Products of ultracompact spaces are ultracompact” + “there exists a free ultrafilter on ω” (UF(ω)) is equivalent to AC in ZFA;2.“Products of ultracompact spaces are ultracompact” does not imply AC in ZF;3.AC is equivalent to each of “products of ultracompact spaces are countably compact”, “products of ultracompact spaces are sequentially limit point compact”, “products of ℵ0-bounded spaces are countably compact”, “products of ℵ0-bounded spaces are sequentially limit point compact”, and “products of ℵ0-bounded spaces are ℵ0-bounded” in ZFA;4.UF(ω) is equivalent to each of “every ultracompact space is sequentially limit point compact” and “every ultracompact T2 space is sequentially limit point compact”;5.The Axiom of Countable Choice (ACω) is equivalent to “every ℵ0-bounded space is countably compact”;6.ACω+UF(ω) is equivalent to each of “every ultracompact space is countably compact”, “products of countably many ultracompact spaces are countably compact”, “products of countably many ultracompact spaces are sequentially limit point compact”, and “products of countably many ℵ0-bounded spaces are ℵ0-bounded”;7.Each of “a T3 space is ultracompact, if and only if, it is ℵ0-bounded” and “a Tychonoff space is ultracompact, if and only if, it is ℵ0-bounded” is equivalent to “every filter on ω can be extended to an ultrafilter on ω” (BPI(ω)), and thus each of the above statements is equivalent to the statement “the Stone space βω of all ultrafilters on ω is compact”. Thus we provide the exact characterizations of the above topological results, the former by Vaughan in “Countably compact and sequentially compact spaces”, and the latter by Bernstein in “A new kind of compactness for topological spaces”, as a specific renowned weak choice principle (which is strictly weaker than the Boolean Prime Ideal Theorem in ZF);8.ACω implies Ginsburg's theorem in “Some results on the countable compactness and pseudocompactness of hyperspaces”, namely “a T2 space X is ultracompact, if and only if, the hyperspace CL(X) (the set of non-empty closed subsets of X with the Vietoris topology) is ultracompact”, which in conjunction with UF(ω) implies the weak choice principle ACfinω (AC restricted to countable families of non-empty finite sets).

On continuous images of ultra-arcs

Any space homeomorphic to one of the standard subcontinua of the Stone-Čech remainder of the real half-line is called an ultra-arc. Alternatively, an ultra-arc may be viewed as an ultracopower of the real unit interval via a free ultrafilter on a countable set. It is known that any continuum of weight ≤ℵ1 is a continuous image of any ultra-arc; in this paper we address the problem of which continua are continuous images under special maps. Here are some of the results we present.•Every nondegenerate locally connected chainable continuum of weight ≤ℵ1 is a co-elementary monotone image of any ultra-arc.•Every nondegenerate chainable metric continuum is a co-existential image of any ultra-arc.•Every chainable continuum of weight ℵ1 is a co-existential image of any ultra-arc whose indexing ultrafilter is a Fubini product of two free ultrafilters.•There is a family of continuum-many topologically distinct nonchainable metric continua, each of which is a co-existential image of any ultra-arc.•A nondegenerate continuum which is either a monotone or a co-existential image of an ultra-arc cannot be aposyndetic–let alone locally connected–without being a generalized arc.

A remark on the ultrapower algebra of the hyperfinite factor

On page 43 in [Adv. in Math. 50 (1983), pp. 27–48] Sorin Popa asked whether the following property holds: If ω \omega is a free ultrafilter on N \mathbb N and R 1 ⊆ R \mathcal {R}_1\subseteq \mathcal {R} is an irreducible inclusion of hyperfinite II 1 _1 factors such that R ′ ∩ R ω ⊆ R 1 ω \mathcal {R}’\cap \mathcal {R}^\omega \subseteq \mathcal {R}^\omega _1 does it follows that R 1 = R \mathcal {R}_1=\mathcal {R} ? In this short note we provide an affirmative answer to this question.

Metrically Ramsey ultrafilters

Given a metric space $(X,d)$, we say that a mapping $\chi: [X]^{2}\longrightarrow\{0.1\}$ is an isometric coloring if $d(x,y)=d(z,t)$ implies $\chi(\{x,y\})=\chi(\{z,t\})$. A free ultrafilter $\mathcal{U}$ on an infinite metric space $(X,d)$ is called metrically Ramsey if, for every isometric coloring $\chi$ of $[X]^{2}$, there is a member $U\in\mathcal{U}$ such that the set $[U]^{2}$ is $\chi$-monochrome. We prove that each infinite ultrametric space $(X,d)$ has a countable subset $Y$ such that each free ultrafilter $\mathcal{U}$ on $X$ satisfying $Y\in\mathcal{U}$ is metrically Ramsey. On the other hand, it is an open question whether every metrically Ramsey ultrafilter on the natural numbers $\mathbb{N}$ with the metric $|x-y|$ is a Ramsey ultrafilter. We prove that every metrically Ramsey ultrafilter $\mathcal{U}$ on $\mathbb{N}$ has a member with no arithmetic progression of length 2, and if $\mathcal{U}$ has a thin member then there is a mapping $f:\mathbb{N}\longrightarrow\omega $ such that $f(\mathcal{U})$ is a Ramsey ultrafilter.

On non-surjective coarse isometries between Banach spaces

Assume that X, Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f: X → Y is a standard coarse isometry. In this paper, we show that ifthen there is a linear isometry U : X → Y so thatwhere is defined byRepresentation properties of coarse isometries in free ultrafilter limits on are also discussed.

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:Given an infinite set X, the Stone space S(X) is ultrafilter compact.For every infinite set X, every countable filterbase of X extends to an ultra-filter iff for every infinite set X, S(X) is countably compact.ω has a free ultrafilter iff every countable, ultrafilter compact space is countably compact.We also show the following:There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ℝ which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ℝ has free ultrafilters but there exists a countable filterbase of ℝ which does not extend to an ultrafilter.

Density measures and additive property

We consider a certain class of normalized positive linear functionals on l∞ which extend the Cesàro mean. We study the set of its extreme points and it turns out to be the set of linear functionals constructed from free ultrafilters on natural numbers N. Also, regarding them as finitely additive measures defined on all subsets of N, which are often called density measures, we study a certain additivity property of such measures being equivalent to the completeness of the Lp-spaces on such measures. Particularly a necessary and sufficient condition for such a density measure to have this property is obtained.

Selective and Ramsey Ultrafilters on G-spaces

Let G be a group, and let X be an infinite transitive G-space. A free ultrafilter U on X is called G-selective if, for any G-invariant partition P of X, either one cell of P is a member of U, or there is a member of U which meets each cell of P in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a G-selective ultrafilter on X. We describe all G-spaces X such that each free ultrafilter on X is G-selective, and we prove that a free ultrafilter U on ω is selective if and only if U is G-selective with respect to the action of any countable group G of permutations of ω. A free ultrafilter U on X is called G-Ramsey if, for any G-invariant coloring χ:[X]2→{0,1}, there is U∈U such that [U]2 is χ-monochromatic. We show that each G-Ramsey ultrafilter on X is G-selective. Additional theorems give a lot of examples of ultrafilters on Z that are Z-selective but not Z-Ramsey.