Countable Dense Set Research Articles

Countable dense sets of dendrites

For each separable space X, we consider the family of all countable dense sets of X modulo an equivalence relation defined as follows: we say that M∼N if and only if there is a homeomorphism h:X→X such that h[M]=N. The countable dense homogeneity degree of X is defined as the cardinality of the set of the equivalence classes under the relation ∼. In case such degree equals 1, the space is said to be countable dense homogeneous. Countable dense homogeneity has become a classical concept and has been studied with interest in the last decades. In this paper we present a full characterization of the countable dense homogeneity degree for the class of dendrites.

Some dense sets of the Tychonoff cube Ic

By Hewitt–Marczewski–Pondiczery theorem (see [3]) the Tychonoff product of 2ω many separable spaces ▪ is separable.The problem of the existence of countable dense sets in a product of separable spaces ▪ with additional properties is very important.We prove that in the Tychonoff cube Ic there are dense sets such that all their countable subsets have properties, which entail, in particular, that these dense sets contain no convergent sequences.

Flexibility of the Pressure Function

We study the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics, since they correspond to phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We prove that in a setting of one-dimensional compact symbolic systems these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. In fact, we establish a multidimensional version of this result. As a consequence, we obtain that for a continuous observable the phase transitions can occur at a countable dense set of temperature values. We go further and show that one can vary the cardinality of the set of ergodic equilibrium states as a function of the parameter to be any number, finite or infinite.

The integrated density of states of the 1D discrete Anderson–Bernoulli model at rational energies

We show that there is a countable dense set of energies at which the integrated density of states of the 1D discrete Anderson–Bernoulli model can be given explicitly and does not depend on the disorder parameter, provided that the latter is above an energy-dependent threshold.

On the cardinality of separable pseudoradial spaces

The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is in ZFC a regular (or just Hausdorff) SP space of cardinality >c. While this question is left open, we establish a number of non-trivial results that we list below.•It is consistent with MA+c=ℵ2 that there is a countably tight and compact SP space of cardinality 2c.•If κ is a measurable cardinal then in the forcing extension obtained by adding κ many Cohen reals, every countably tight regular SP space has cardinality at most c.•If κ>ℵ1 Cohen reals are added to a model of GCH then in the extension every pseudocompact SP space with a countable dense set of isolated points has cardinality at most c.•If c≤ℵ2 then there is a 0-dimensional SP space with a countable dense set of isolated points that has cardinality greater than c.

On Strong Continuity of Weak Solutions to the Compressible Euler System

Let \({\mathcal {S}} = \{ \tau _n \}_{n=1}^\infty \subset (0,T)\) be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions that are not strongly continuous at each \(\tau _n\), \(n=1,2,\dots \). The proof is based on a refined version of the oscillatory lemma of De Lellis and Székelyhidi with coefficients that may be discontinuous on a set of zero Lebesgue measure.

On Measurability of C[0, 1]: Countable Dense Set Approach

In V. Boonpogkrong, On measurability of [Formula: see text], Thai J. Math. (accepted) we have proved that [Formula: see text] is measurable in a setting of a dyadic Henstock integral, where dimension sets are subsets of the dyadic rational. In this paper, we replace “dyadic rational” by “countable dense set” and use a setting of Henstock integral, where dimension sets are subsets of this fixed countable dense set. The collection of measurable sets in this new setting is smaller than that in the old setting. We still can prove that [Formula: see text] is measurable.

Generalized 2-Microlocal Frontier Prescription

The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in $$\mathbb {R}^2$$ , provides a useful way to classify pointwise singularity. In this paper we characterize all functions whose 2-microlocal frontier at a given point $$x_0$$ is a given line. Further, for a general concave downward curve, we obtain a large family of functions (or distributions) for which the 2-microlocal frontier is the given curve. This family contains—as special cases—the constructions given in Meyer (CRM monograph series, 1998), Guiheneuf et al. (ACHA 5(4):487–492, 1998) and Levy Vehel et al. (Proc Symp Pure Math 72:319–334, 2004). Moreover, following Levy Vehel et al. (2004), we extend our results to the prescription on a countable dense set.

Strongly sequentially separable function spaces, via selection principles

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continuous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.

Differentiability of the functions of the generalized Takagi Class

We introduce a generalization of the Takagi Class, considering an arbitrary countable dense set instead of the dyadic numbers that appear in the original Takagi function. This generalization contains many of the previous generalizations. We study the differentiation of the functions belonging to this Class, as well as the measure of the set of points of differentiability characterizing them through properties of the weights sequence. Finally, we prove that the Class is a Banach space isometric to $$\ell ^1$$.

Universal Entire Functions That Define Order Isomorphisms of Countable Real Sets

AbstractIn 1895, Cantor showed that between every two countable dense real sets, there is an order isomorphism. In fact, there is always such an order isomorphism that is the restriction of a universal entire function.

Geometric random graphs and Rado sets in sequence spaces

We consider a random geometric graph model, where pairs of vertices are points in a metric space and edges are formed independently with fixed probability p between pairs within threshold distance 1. A countable dense set in a metric space is Rado if this random model gives, with probability 1, a graph that is unique up to isomorphism. In earlier work, the first two authors proved that, in finite dimensional spaces Rn equipped with the ℓ∞ norm, all countable dense sets satisfying a mild non-integrality condition are Rado. In this paper we extend this result to two infinite-dimensional spaces: c and c0. If the underlying metric space is a separable Banach space, then we show in some cases that we can almost surely recover the Banach space from such a geometric random graph. More precisely, we show that in the sequence spaces c and c0, for measures μ satisfying certain conditions, μN-almost all countable sets are Rado. Moreover, with probability 1, in c as in c0, all graphs obtained from the random geometric model with a randomly chosen dense countable vertex set are isomorphic to each other. Finally, we show that representatives of the isomorphism classes obtained in this way from c and c0 are non-isomorphic to each other, and also non-isomorphic to their counterparts obtained from finite dimensional spaces.

Space of Continuous Set-Valued Mappings with Closed Unbounded Values

We consider a space of continuous set-valued mappings defined on a locally compact space T with a countable base. The values of these mappings are closed (not necessarily bounded) sets in a metric space (X, d(·)) whose closed balls are compact. The space (X, d(·)) is locally compact and separable. Let Y be a countable dense set in X. The distance ρ(A,B) between sets A and B belonging to the family CL(X) of all nonempty closed subsets of X is defined as follows: $$\rho (A, B) = \sum_{i=1}^\infty\frac{1}{2^i}\frac{|d(y_i, A) - d(y_i, B)|}{1+|d(y_i, A) - d(y_i, B)|},$$ where d(yi,A) is the distance from the point yi ∈ Y to the set A. This distance is independent of the choice of the set Y, and the function ρ(A,B) is a metric on the space CL(X). The convergence of a sequence of sets An, n ≥ 1, in the metric space (CL(X), ρ(·)) is equivalent to the Kuratowski convergence of this sequence. We prove the completeness and separability of the space (CL(X), ρ(·)) and give necessary and sufficient conditions for the compactness of sets in this space. The space C(T,CL(X)) of all continuous mappings from T to (CL(X), ρ(·)) is endowed with the topology of uniform convergence on compact sets in T. We prove the completeness and separability of the space C(T,CL(X)) and give necessary and sufficient conditions for the compactness of sets in this space. These results are reformulated for the space C(T,CCL(X)), where T = [0, 1], X is a finite-dimensional Euclidean space, and CCL(X) is the space of all nonempty closed convex sets in X with metric ρ(·). This space plays a crucial role in the study of sweeping processes. We give a counterexample showing the significance of the assumption of compactness of closed balls in X.

On dense subsets of Tychonoff products of T1-spaces

We consider dense subsets of Zc=∏α∈2ωZα, where Z=Zα is a separable not single point T1-space.We construct in Zc, |Z|≥ω, (Theorem 4.1) a countable dense set Q⊆Z such that every countable subset of Q contains a countable subset, which can be project on “many” subsets of Z.From this theorem follow some facts.Theorem 4.2 states that in the product Zc of a not single point separable T1-space Z there is a countable dense set which contains no non-trivial convergent in Zc sequences.The existence of such set in the product Ic, where I=[0,1] was proved [13] by W. H. Priestley.In [9] we proved that such set exists in a product of 2ω separable not single point T2-spaces.Theorem 4.4 states that if Z is a separable not countably compact T1-space, then there is a countable dense subset Q⊆Zc, satisfying the following condition: if E⊆Q is a countable set and E converges to a set F⊆Zc, then |E∖F|<ω.

Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus

We show that for every sequence $(n_i)$, where each $n_i$ is either an integer greater than 1 or is $\infty$, there exists a simply connected open 3-manifold $M$ with a countable dense set of ends $\{e_i\}$ so that, for every $i$, the genus of end $e_i$ is equal to $n_i$. In addition, the genus of the ends not in the dense set is shown to be less than or equal to 2. These simply connected 3-manifolds are constructed as the complements of certain Cantor sets in $S^3$. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus.

Hyperuniformity of quasicrystals

Hyperuniform systems, which include crystals, quasicrystals and special disordered systems, have attracted considerable recent attention, but rigorous analyses of the hyperuniformity of quasicrystals have been lacking because the support of the spectral intensity is dense and discontinuous. We employ the integrated spectral intensity, $Z(k)$, to quantitatively characterize the hyperuniformity of quasicrystalline point sets generated by projection methods. The scaling of $Z(k)$ as $k$ tends to zero is computed for one-dimensional quasicrystals and shown to be consistent with independent calculations of the variance, $\sigma^2(R)$, in the number of points contained in an interval of length $2R$. We find that one-dimensional quasicrystals produced by projection from a two-dimensional lattice onto a line of slope $1/\tau$ fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, $Z(k) \sim k^4$; for all others, $Z(k)\sim k^2$. This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.

Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.

Addendum to “Nearly Countable Dense Homogeneous Spaces”

This paper provides an addendum to M. Hrusak and J. van Mill “Nearly Countable Dense Homogeneous Spaces.” Canad. J. Math., published online 2013-03-08, http://dx.doi.org/10.4153/ CJM-2013-006-8. It was brought to our attention by Su Gao that the proof of Theorem 5.2 in our paper is incomplete. We are indebted to him for this observation. The aim of this note is to correct this. Theorem 5.2 Let G be a closed subgroup of S∞ and let κ be the number of orbits for the canonical action G × 2N → 2N. Then there is an action of a Polish group H on X = N× [0, 1) such that X has κ H-types of countable dense sets. Proof Let G act on X in the following natural way: (g, (n, t)) 7−→ (g(n), t) for g ∈ G, n ∈ N, t ∈ [0, 1). Put F = { f ∈H (X) : (∀ n ∈ N)( f (n, 0) = (n, 0)) } . Then F is a closed normal subgroup of H (X) and hence is Polish. Moreover, for any two countable dense subsets D and E of N × (0, 1) there exists f ∈ F such that f (D) = E. Treat G also as subgroup of H (X). The Polish semi-direct product group H = G o F acts on X as follows: ((g, f ), x) 7→ ( f ◦ g)(x) for f ∈ F, g ∈ G, x ∈ X. Note that topologically, H = G o F is G× F, but its group operation ∗ is given by (g1, f1) ∗ (g2, f2) = (g1g2, f1g1 f2g 1 ). A typical countable dense subset of X has the form D ∪ A, where D is a countable dense subset of N× (0, 1), and A ⊆ N× {0}. By identifying P(N× {0}) and 2N in the standard way, it is clear that we get what we want. Centro de Ciencias Matematicas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacan, 58089, Mexico e-mail: michael@matmor.unam.mx Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands e-mail: j.van.mill@vu.nl Received by the editors October 28, 2013. Published electronically November 7, 2013. The first author was supported by a PAPIIT grant IN 102311 and CONACyT grant 177758. The second author is pleased to thank the Centro de Ciencias Matematicas at Morelia for generous hospitality and support. AMS subject classification: 54H05, 03E15, 54E50.

Nearly Countable Dense Homogeneous Spaces

AbstractWe study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X ∖ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or c many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.

On dense subsets of Tychonoff products

The Hewitt–Marczewski–Pondiczery theorem states that if X=∏α∈AXα is the Tychonoff product of spaces, where d(Xα)≤τ≥ω for all α∈A and |A|≤2τ, then d(X)≤τ.We prove that if ∏α∈2ωXα is the Tychonoff product of 2ω many spaces and d(Xα)=ω for all α∈2ω, then there is the countable dense set Q⊆∏α∈2ωXα such that1.Q=⋃{Qj:j∈ω}, |Qj|<ω for all j∈ω and Qi∩Qj=∅ if i≠j;2.if C⊆ω is an infinite subset, then the set ⋃{Qj:j∈C} is dense in ∏α∈2ωXα;3.if for a set F⊆Q there is m0∈ω such that |Qk∩F|≤m0 for all k∈ω, then the set Q∖F is dense in ∏α∈2ωXα.