Countable Product Research Articles

Dense products in fundamental groupoids

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid $\Pi_1(X)$ has well-defined dense products if and only if $X$ admits a generalized universal covering space.

Embeddings for infinite-dimensional integration and [formula omitted]-approximation with increasing smoothness

We study integration and L2-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

Groups of line and circle homeomorphisms. Criteria for almost nilpotency

For finitely-generated groups of line and circle homeomorphisms a criterion for their being almost nilpotent is established in terms of free two-generator subsemigroups and the condition of maximality. Previously the author found a criterion for almost nilpotency stated in terms of free two-generator subsemigroups for finitely generated groups of line and circle homeomorphisms that are -smooth and mutually transversal. In addition, for groups of diffeomorphisms, structure theorems were established and a number of characteristics of such groups were proved to be typical. It was also shown that, in the space of finitely generated groups of -diffeomorphisms with a prescribed number of generators, the set of groups with mutually transversal elements contains a countable intersection of open dense subsets (is residual). Navas has also obtained a criterion for the almost nilpotency of groups of -diffeomorphisms of an interval, where , in terms of free subsemigroups on two generators. Bibliography: 21 titles.

The Distant Graph of the Ring of Integers and Its Representations in the Modular Group

There is presented an infinite class of subgroups of the modular group mathrm{PSL}(2,mathbb {Z}) that serve as Cayley representations of the distant graph of the projective line of integers. They are infinite countable free products of subgroups of mathrm{PSL}(2,mathbb {Z}) isomorphic with mathbb {Z}_2, mathbb {Z}_3 and mathbb {Z} subject to the restriction that the number of copies of mathbb {Z} is 0 or 2. The proof technique is based on a 1–1 correspondence between some involutions iota of mathbb {Z} that fulfill the equation ι(ι(n)-δn)=ι(n+1)+δn+1,δn=±1,δι(n)=δn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\iota (\\iota (n)-\\delta _n)=\\iota (n+1)+\\delta _{n+1},\\quad \\delta _n=\\pm \\,1, \\quad \\delta _{\\iota (n)}=\\delta _n, \\end{aligned}$$\\end{document}and groups from this class.

Some thoughts on countable Lindelöf products

A space X for which Xω is Lindelöf is called powerfully Lindelöf. Extending this term, I call a collection of topological spaces powerfully Lindelöf if the product of each of its countable subcollections is Lindelöf. In 1971 E. Michael showed (CH) that there is no single such class; here I explore what can be said of them. In doing so, I will focus upon some of the techniques used to prove countable products Lindelöf.

On linearly H-closed spaces

We study a subclass of the class of pseudocompact spaces, the linearly H-closed spaces, recently introduced by M. Baillif. We give two new characterizations of these spaces in terms of filters and complete accumulation points of families of open sets. We show that a monotonically normal linearly H-closed space is compact and that linear H-closedness is preserved under a product with an H-closed space. Additionally, a condition is given in order that the property be preserved under countable products.

Unitary representations of infinite wreath products

Using C∗-algebraic techniques and especially AF-algebras, we present a complete classification of the continuous unitary representations for a class of infinite wreath product groups. These nonlocally compact groups are realized by a topological completion of the semidirect product of the countably infinite symmetric group acting on the countable direct product of a finite Abelian group.

Strong Cosmic Censorship in Orthogonal Bianchi Class B Perfect Fluids and Vacuum Models

The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein’s field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curvature invariants such as the Kretschmann scalar is a generic property. The only spacetimes where this scalar remains bounded exhibit local rotational symmetry or are of plane wave equilibrium type. We further investigate the qualitative behaviour of solutions towards the initial singularity. To this end, we work in the expansion-normalised variables introduced by Hewitt–Wainwright and show that a set of full measure, which is also a countable intersection of open and dense sets in the state space, yields convergence to a specific subarc of the Kasner parabola. We further give an explicit construction enabling the translation between these variables and geometric initial data to Einstein’s equations.

Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Generalising a construction of Falconer, we consider classes of G δ -subsets of ℝ d with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.

The G-connected property and G-topological groups

In this paper, we discuss some properties of of G-hull, G-kernel and G-connectedness, and extend some results of [28]. In particular, we prove that the G-connectedness are preserved by countable product. Moreover, we introduce the concept of G-topological group, and prove that a G-topological group is a G-topology under the assumption of the regular method preserving the subsequence.

On sets where $\operatorname{lip} f$ is finite

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called little lip function $\operatorname{lip} f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r} \frac{|f(y)-f(x)|}{r}. \end{equation*} We show that if $f$ is continuous on $\mathbb{R}$, then the set where $\operatorname{lip} f$ is infinite is a countable union of a countable intersection of closed sets (that is an $F_{\sigma \delta}$ set). On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\operatorname{lip} f$ is infinite exactly on $E$. A further result is that for the typical continuous function $f$ on the real line $\operatorname{lip} f$ vanishes almost everywhere.

A random weak ergodic property of infinite products of operators in metric spaces

ABSTRACTWe study the random weak ergodic property of infinite products of mappings acting on complete metric spaces. Our results describe an aspect of the asymptotic behaviour of random infinite products of such mappings. More precisely, we show that in appropriate spaces of sequences of operators there exists a subset, which is a countable intersection of open and everywhere dense sets, such that each sequence belonging to this subset has the random weak ergodic property. Then we show that several known results in the literature can be deduced from our general result.

$$\infty $$ ∞ -Constructible subsemigroups of $$M_2(\mathbb {C})$$ M 2 ( C )

A description of all subsemigroups of $$M_2(\mathbb {C})$$ which are given by a countable intersection of constructible sets is obtained. Furthermore, it is shown that they are intersections of constructible semigroups.

An explicit compact universal space for real flows

The Kakutani–Bebutov Theorem (1968) states that any compact metric real flow whose fixed point set is homeomorphic to a subset of R embeds into the Bebutov flow, the R-shift on C(R,[0,1]). An interesting fact is that this universal space is a function space. However, it is not compact, nor locally compact. We construct an explicit countable product of compact subspaces of the Bebutov flow which is a universal space for all compact metric real flows, with no restriction; namely, into which any compact metric real flow embeds. The result is compared to previously known universal spaces.

An Infinite Natural Product

Abstract We study a countably infinite iteration of the natural product between ordinals.We present an “effective” way to compute this countable natural product; in the non trivial cases the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product; this characterization merges in a nontrivial way a theorem by Carruth describing the natural product of two ordinals and a known description of the ordinal product of a possibly infinite sequence of ordinals.

Small sets containing any pattern

AbstractGiven any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

The abelianization of inverse limits of groups

The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all small colimits. In this paper we investigate the relation between the abelianization of a limit of groups and the limit of their abelianizations. We show that if T is a countable directed poset and G: T → Grp is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$Ab(\mathop {lim}\limits_{t \in T} {G_t}) \to \mathop {lim}\limits_{t \in T} Ab({G_t})$$ is surjective, and its kernel is a cotorsion group. In the special case of a countable product of groups, we show that the Ulm length of the kernel does not exceed ℵ1.

A Lindelöfication

An almost real maximal ideal M of C(X) is a maximal ideal that is either fixed or Z[M] contains a free z-filter which is closed under countable intersection. Using these maximal ideals, we first construct a space λX which is weakly Lindelöf containing X as a dense subspace on which every f∈C(X) with Lindelöf cozero-set can be extended (when this is the case, we say that X is CL-embedded). Next, using this space, we present the largest Lindelöf subspace ΛX of βX in which X is CL-embedded. If X is locally Lindelöf, λX coincides with ΛX and it turns out in this case that ΛX(=λX) is the smallest Lindelöf subspace of βX with compact remainder. Using the structure of ΛX, we also give the smallest realcompact subspace ϒX of βX with compact remainder. Finally the relations between the spaces υX, λX, ϒX, ΛX and βX are investigated and we apply the structures of these spaces to characterize some intersections of free maximal ideals of C(X).

Quasi-metrizability of products in ZF and equivalences of CUT(fin)

It is proved in ZF that if a collection of (quasi)-metric spaces is indexed by a countable union of finite sets, then the product of this collection is (quasi)-metrizable, while it is independent of ZF that every countable product of metrizable spaces is quasi-metrizable. It is also proved that if J is a non-empty set, while a space X, consisting of at least two points, is equipped with the co-finite topology, then the product XJ is quasi-metrizable if and only if both X and J are countable unions of finite sets. Several equivalences of CUT(fin) are deduced. It is shown that, in every model of ZF+¬CUT(fin), for an uncountable set J, the space NJ can be normal (even metrizable), a Cantor cube can be simultaneously metrizable, not second-countable and non-compact.

Quasi-invariance of countable products of Cauchy measures under non-unitary dilations

Consider an infinite sequence $(U_n)_{n\in \mathbb{N} }$ of independent Cauchy random variables, defined by a sequence $(\delta _n)_{n\in \mathbb{N} }$ of location parameters and a sequence $(\gamma _n)_{n\in \mathbb{N} }$ of scale parameters. Let $(W_n)_{n\in \mathbb{N} }$ be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence $(\sigma _n\gamma _n)_{n\in \mathbb{N} }$ of scale parameters, with $\sigma _n\neq 0$ for all $n\in \mathbb{N} $. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of $(U_n)_{n\in \mathbb{N} }$ and $(W_n)_{n\in \mathbb{N} }$ are equivalent if and only if the sequence $(\left \vert{\sigma _n} \right \vert -1)_{n\in \mathbb{N} }$ is square-summable.