Urysohn Space Research Articles

What would the rational Urysohn space and the random graph look like if they were uncountable?

What would the rational Urysohn space and the random graph look like if they were uncountable?

Computability Theory on Polish Metric Spaces

AbstractComputability theoretic aspects of Polish metric spaces are studied by adapting notions and methods of computable structure theory. In this dissertation, we mainly investigate index sets and classification problems for computably presentable Polish metric spaces. We find the complexity of a number of index sets, isomorphism problems, and embedding problems for computably presentable metric spaces. We also provide several computable structure theory results related to some classical Polish metric spaces such as the Urysohn space $\mathbb {U}$ , the Cantor space $2^{\mathbb {N}}$ , the Baire space $\mathbb {N}^{\mathbb {N}}$ , and spaces of continuous functions.Abstract prepared by Teerawat Thewmorakot.E-mail: teerawat.thew@hotmail.com

Characterizations of Some Topological Spaces

This paper is concerned with the concepts of some topological spaces. Firstly, we introduce the notions of δs(Λ, p)-open sets. Some properties concerning δs(Λ, p)-open sets are discussed. Secondly, the concept of s(Λ, p)-connected spaces is introduced. Moreover, we give several characterizations of s(Λ, p)-connected spaces by utilizing δs(Λ, p)-open sets. Thirdly, we apply the notion of s(Λ, p)-open sets to present and study new classes of spaces called s(Λ, p)-regular spaces and s(Λ, p)-normal spaces. Especially, some characterizations of s(Λ, p)-regular spaces and s(Λ, p)-normal spaces are established. Fourthly, we introduce and investigate the concepts of s(Λ, p)-T2 spaces and s(Λ, p)-Urysohn spaces. Finally, the notion of S(Λ, p)-closed spaces is studied. Basic properties and characterizations of S(Λ, p)-closed spaces are considered.

Selective separability properties of Fréchet–Urysohn spaces and their products

Selective separability properties of Fréchet–Urysohn spaces and their products

Cardinal inequalities with Shanin number and $\pi$-character

It is proved that, under GCH, for any Hausdorff space $X$, we have the inequality $|X| \leq sh(X)^{\pi\chi(X) \cdot \psi_c(X)}$ and therefore $|X\leq sh(X)^{\chi(X)}$; here $sh(X)=\min\{\kappa\geq \omega: \kappa^+$ is a caliber of $X\}$ is the Shanin number of $X$. If $X$ is regular, and GCH holds, then $d(X) \leq sh(X)^{t(X)\cdot \pi\chi(X)}$. We also establish, in ZFC, that $|X| \leq wL(X)^{\CL\Delta(X)\cdot 2^{\pi\chi(X)}}$ whenever $X$ is a Urysohn space.

On Sw*- Regular Spaces

The purpose of this paper is to present and investigate a new class of topological spaces known as Sw*-regular spaces, by utilizing the concept of Sw-regular sets and some of its properties. which is introduced in 2009 by L. S. Abddullah and A. B. Khalaf [1], the new class is properly contained in S*- regular space [2], [3], means that Sw*- regular spaces is a stronger form to the space S*- regular. Several characterizations, properties and relationships of Sw*- regular space with other spaces such as, Sw-compact, extremally disconnected, regular, semi-regular, Sw-T2 and Urysohn spaces has been studied. Furthermore, several properties of Sw*- regular spaces with some functions such as, continuous, strongly continuous, open, clopen and Somewhat open functions are also explored. In addition we investigate that Sw*- regular space has a topological property, while it has not a hereditary property, only by adding certain conditions such as, a subspace is open or, if the subspace of an Sw*-regular submaximal space is preopen, then it becomes an Sw*- regular.

Homogeneous actions on Urysohn spaces

We show that many countable groups acting on trees, including free products of infinitely countable groups and surface groups, are isomorphic to dense subgroups of isometry groups of bounded Urysohn spaces. This extends previous results of the first and t

Cardinal estimates involving the weak Lindel\xf6f game

We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game G^{omega _1}_1({mathcal {O}}, {mathcal {O}}_D) (the weak Lindelöf game of length omega _1) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game G^{omega _1}_{fin}({mathcal {O}}, {mathcal {O}}_D), providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length omega _1.

Dense locally finite subgroups of automorphism groups of ultraextensive spaces

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal Kn-free graph for all n≥3. Furthermore, we show that finite group actions on finite metric spaces or some finite relational structures form a Fraïssé class, where Hall's group appears as the acting group of the Fraïssé limit. We also embed continuum many non-isomorphic universal countable locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn Δ-metric spaces in terms of the distance value set Δ.

Fibrewise totally perfect mapping

The main purpose of this paper is to introduce a some concepts in fibrewise totally topological space which are called fibrewise totally mapping, fiberwise totally closed mapping, fibrewise weakly totally closed mapping, fibrewise totally perfect mapping fibrewise almost totally perfect mapping. Also the concepts as totally adherent point, filter, filter base, totally converges to a subset, totally directed toward a set, totally rigid, totally-H-set, totally Urysohn space, locally totally-QHC totally topological space are introduced and the main concept in this paper is fibrewise totally perfect mapping in totally topological space. Several theorem and characterizations concerning with these concepts are studied.

Simplicity of the automorphism groups of generalised metric spaces

Tent and Ziegler proved that the automorphism group of the Urysohn sphere is simple and that the automorphism group of the Urysohn space is simple modulo bounded automorphisms. A key component of their proof is the definition of a stationary independence relation (SIR). In this paper we prove that the existence of a SIR satisfying some extra axioms is enough to prove simplicity of the automorphism group of a countable structure. The extra axioms are chosen with applications in mind, namely homogeneous structures which admit a “metric-like amalgamation”, for example all primitive 3-constrained metrically homogeneous graphs of finite diameter from Cherlin's list.

Simplicity of the automorphism groups of order and tournament expansions of homogeneous structures

We define the notions of a free fusion of structures and a weakly stationary independence relation and use them to prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like the rational bounded Urysohn space, the random graph, and the random poset.

Maximal equivariant compactification of the Urysohn spaces and other metric structures

We study isometric G-spaces and the question of when their maximal equivariant compactification is the Gromov compactification (meaning that it coincides with the compactification generated by the distance functions to points). Answering questions of Pestov, we show that this is the case for the Urysohn sphere and related spaces, but not for the unit sphere of the Gurarij space.We show that the maximal equivariant compactification of a separably categorical metric structure M under the action of its automorphism group can be identified with the space S1(M) of 1-types over M, and is in particular metrizable. This provides a unified understanding of the previous and other examples. In particular, the maximal equivariant compactifications of the spheres of the Gurarij space and of the Lp spaces are metrizable.We also prove a uniform version of Effros' Theorem for isometric actions of Roelcke precompact Polish groups.

A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

Novel Idea of Gδ-α-Locally Continuous Functions

The new concepts of a neutrosophic Gδ set and neutrosophic Gδ-α-locally closed sets are introduced. Also, a neutrosophic εGδ-α-locally quasi neighborhood, neutrosophic Gδ-α-locally continuous function, neutrosophic Gδ-α-local T2 space, neutrosophic Gδ-α-local Urysohn space, neutrosophic Gδ-α-local connected space, and neutrosophic Gδ-α-local compact space are discussed and some interesting properties are established.

SUPRA CO BT URYSOHN SPACE IN SUPRA TOPOLOGICAL SPACES

In this article, the author interpolate prolific notion of co $bT^{\mu}$ graph and co $bT^{\mu}$ urysohn space by utilizing the concept of bT$^{\mu}$ set connected functions in supra topological spaces.

On cellular-compact and related spaces

We solve a problem posed by Tkachuk and Wilson, Question 5.10 in [16], on whether every first countable cellular-compact space is weakly Lindelöf. We actually obtain a stronger result and, as a by-product of it, we present a somewhat different proof of Tkachuk and Wilson theorem on the cardinality of a first countable cellular-compact space, valid for the wider class of Urysohn spaces. Moreover, our result holds for a class of spaces in between cellular-compact and cellular-Lindelöf. We conclude with some comments on the cardinality of a weakly linearly Lindelöf space.

Generic representations of countable groups

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups Γ \Gamma in Polish groups G G , i.e., elements in the Polish space R e p ( Γ , G ) \mathrm {Rep}(\Gamma ,G) of all representations of Γ \Gamma in G G whose orbits under the conjugation action of G G on R e p ( Γ , G ) \mathrm {Rep}(\Gamma ,G) are comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or K n K_n -free graphs, and we show its connections with Ribes–Zalesskii-like properties of the acting groups. We prove that Z \mathbb {Z} has a generic representation in the automorphism group of the random tournament (i.e., there is a comeager conjugacy class in this group). We formulate a Ribes–Zalesskii-like condition on a group that guarantees finite approximability of its actions on tournaments. We also provide a simpler proof of a result of Glasner, Kitroser, and Melleray characterizing groups with a generic permutation representation. We also investigate representations of infinite groups Γ \Gamma in automorphism groups of metric structures such as the isometry group Iso ( U ) \mbox {Iso}(\mathbb {U}) of the Urysohn space, isometry group Iso ( U 1 ) \mbox {Iso}(\mathbb {U}_1) of the Urysohn sphere, or the linear isometry group LIso ( G ) \mbox {LIso}(\mathbb {G}) of the Gurarii space. We show that the conjugation action of Iso ( U ) \mbox {Iso}(\mathbb {U}) on R e p ( Γ , Iso ( U ) ) \mathrm {Rep}(\Gamma ,\mbox {Iso}(\mathbb {U})) is generically turbulent, answering a question of Kechris and Rosendal.

On cardinality bounds for $$\theta^n$$-Urysohn spaces

We introduce the class of $$\theta^n$$ -Urysohn spaces and the $$n$$ - $$\theta$$ -closure operator. $$\theta^n$$ -Urysohn spaces generalize the notion of a Urysohn space and we consider their relationship with S(n)-spaces, studied in [9], [14] and [18]. We estabilish bounds on the cardinality of these spaces and cardinality bounds if the space is additionally homogeneous.

More cardinal inequalities in T1/Urysohn spaces

In this paper we provide a partial positive answer to the question 2 in [8]: Do we have |X|≤2L(X)Fa(X)ψ(X), for every T2–space X? Precisely, in Theorem 2.6 we show that |X|≤2L(X)Fa(X)ψθ(X) for Urysohn spaces. Even more, we use cardinal functions recently introduced by Basile, Bonanzinga and Carlson in [4] to generalize some cardinal inequalities; among others, in the Theorem 2.8, we provide a result related with the well known inequality of Hajnal and Juhász: |X|≤2c(X)χ(X); and in the Theorem 2.10, we generalize the inequality |X|≤2wLc(X)χ(X), for any Urysohn space X; due to Ofelia Alas ([1]). Some proofs of the theorems provided in this paper use the elementary submodels technique.