Metric Spaces Research Articles

Selective properties of fuzzy 2-metric spaces

We introduce and study some selective covering properties in fuzzy 2-metric spaces. These properties are related to the classical covering properties of Menger, Hurewicz and Rothberger which are well known in selection principles theory.

On the Development of Nonlinear Operator Theory

Basic results for nonlinear operators are given. These results include nonlinear versions of the classical uniform boundedness theorem and Hahn-Banach theorem. Further, mappings of a metrizable space into another normed space can belong to some normed spaces if one defines suitable norms.

Parametric 2-metric spaces and some fixed point results

Parametric 2-metric spaces and some fixed point results

Some multi-valued contraction theorems on \(\mathcal{H}\)-cone metric

In this paper we define some new type of multi-valued contraction maps on \(\mathcal{H}\)-cone metric and proved some fixed point and common fixed point theorems in the setting of cone metric spaces.

Fragmentability of open sets and topological games

We give internal and external characterizations of the topological spaces X in which there exists a metric d (not necessarily generating the topology of X) which fragments the nonempty open sets. I.e. every nonempty open subset contains open subsets of arbitrarily small d-diameter. One of the characterizations says that every such space X is the union of two disjoint sets X1 and X2 where X1 is of the first Baire category in X and X2 admits a continuous one-to-one mapping into a metrizable space. An external characterization is obtained via the existence of a winning strategy for one of the players in a topological game similar to the Banach-Mazur game. An example of a space X is exhibited for which neither of the players in this game has a winning strategy.Of special interest is the case when the metric d which fragments the nonempty open sets of X is complete and generates a topology finer than the topology of X. This happens if and only if X contains a dense completely metrizable subset.

Real-valued functions and some related spaces II

In [15], we presented characterizations of some topological spaces such as γ-spaces, semi-metrizable spaces, Nagata spaces and metrizable spaces with real-valued functions. In this paper, we shall continue with the study. Some other forms of characterizations of the topological spaces mentioned above are obtained.

The continuity of Darboux injections between manifolds

We prove that an injective map f:X→Y between metrizable spaces X,Y is continuous if for every connected subset C⊂X the image f(C) is connected and one of the following conditions is satisfied:•Y is a 1-manifold and X is compact and connected;•Y is a 2-manifold and X is a closed 2-manifold;•Y is a 3-manifold and X is a rational homology 3-sphere. This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow.

Existence of quasiequilibria in metric vector spaces

This paper provides sufficient conditions for the existence of solutions for quasiequilibrium problems and generalized game problems in the setting of infinite-dimensional metrizable spaces. To this purpose, we prove a modified version of a selection theorem due to Michael [15] by exploiting the fact that any compact set in a metric space is both complete and separable. Thereafter, by a fixed point technique which is based on the notion of inside point of a convex set, we provide some existence results without requiring the upper semicontinuity and the closed-valuedness of the feasibility maps.

Dunford–Pettis type properties and the Grothendieck property for function spaces

For a Tychonoff space X, let $$C_k(X)$$ and $$C_p(X)$$ be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that $$C_k(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that $$C_k(X)$$ has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space $$[0,\kappa )$$ for some ordinal $$\kappa $$ , or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then $$C_k(X)$$ has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that $$C_p(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and $$C_p(X) $$ has the Grothendieck property if and only if every functionally bounded subset of X is finite.

Characterization of metrizable Esakia spaces via some forbidden configurations

By Priestley duality, each bounded distributive lattice is represented as the lattice of clopen upsets of a Priestley space, and by Esakia duality, each Heyting algebra is represented as the lattice of clopen upsets of an Esakia space. Esakia spaces are those Priestley spaces that satisfy the additional condition that the downset of each clopen is clopen. We show that in the metrizable case Esakia spaces can be singled out by forbidding three simple configurations. Since metrizability yields that the corresponding lattice of clopen upsets is countable, this provides a characterization of countable Heyting algebras. We show that this characterization no longer holds in the uncountable case. Our results have analogues for co-Heyting algebras and bi-Heyting algebras, and they easily generalize to the setting of p-algebras.

Supercompact minus compact is super

According to a folklore characterization of supercompact spaces, a compact Hausdorff space is supercompact if and only if it has a binary closed k-network. This characterization suggests to call a topological space super if it has a binary closed k-network N. The binarity of N means that every linked subfamily of N is centered. Therefore, a Hausdorff space is supercompact if and only if it is super and compact. In this paper we prove that the class of super spaces contains all GO-spaces, all supercompact spaces, all metrizable spaces, and all collectionwise normal ℵ-spaces. Moreover, the class of super spaces is closed under taking Tychonoff products and discretely dense sets in Tychonoff products. The superness of metrizable spaces implies that each compact metrizable space is supercompact, which was first proved by Strok and Szymański (1975) and then reproved by Mills (1979), van Douwen (1981), and Dȩbski (1984).

Zero-dimensional CDH spaces with a dense completely metrizable subset

Given a separable metrizable h-homogeneous space X, it is proved that X is a \(\textsf{CDH}\) space with a dense completely metrizable subspace if and only if every countable subset of X is included in a Polish subspace of X. We also show that every such space is \(\omega-\textsf{CDH}\).

Locally convex properties of free locally convex spaces

Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent: (i) L(X) is ℓ∞-barrelled, (ii) L(X) is ℓ∞-quasibarrelled, (iii) L(X) is c0-barrelled, (iv) L(X) is ℵ0-quasibarrelled, and (v) X is a P-space. If X is a non-discrete metrizable space, then L(X) is c0-quasibarrelled but it is neither c0-barrelled nor ℓ∞-quasibarrelled. We prove that L(X) is a (DF)-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF)-space but is not a c0-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df)-space but is not a quasi-(DF)-space. If X is a μ-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford–Pettis property for every Tychonoff space X. If X is a sequential space and a μ-space (for example, metrizable), then L(X) has the sequential Dunford–Pettis property iff X is discrete.

DISCRETE METRIC SPACES: STRUCTURE, ENUMERATION, AND 0-1 LAWS

AbstractFix an integer $r \ge 3$. We consider metric spaces on n points such that the distance between any two points lies in $\left\{ {1, \ldots ,r} \right\}$. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is $\left\lceil {{{r + 1} \over 2}} \right\rceil ^{\left( {\matrix{ n \cr 2 \cr } } \right) + o\left( {n^2 } \right)} .$Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our structural characterization is more precise and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when r is even, we improve the error term above from $o\left( {n^2 } \right)$ to $o\left( 1 \right)$, and also show a labeled first-order 0-1 law in the language ${\cal L}_r $, consisting of r binary relations, one for each element of $[r]$ . In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\left\{ {r/2, \ldots ,r} \right\}$.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws.

Compactifiable classes of compacta

We introduce the notion of compactifiable classes – these are classes of metrizable compact spaces that can be up to homeomorphic copies “disjointly combined” into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions – we give several characterizations, consider preservation under various constructions, and raise several questions.

Cp-spaces dominated by metrizable topologies

Following Christensen [11] a metrizable space X is σ-compact if and only if Cp(X) is analytic, i.e. it is a continuous image of the Polish space NN. By Michael [26] the space Cp(X) is a continuous image of a separable metric space if and only if X is cosmic, and Cp(X) is an ℵ0-space if and only if X is countable. We show here that, in parallel manner, Christensen-Calbrix and Michael results may be characterized as follows: For a metrizable space X the space Cp(X) is analytic if and only if Cp(X) is a continuous linear image of a separable metrizable locally convex space, and a Tychonoff space X is a submetrizable hemicompact kR-space if and only if Cp(X) is a continuous linear image of a metrizable and complete separable locally convex space. Applications are provided.

Domain-complete and LCS-complete Spaces

We study Gδ subspaces of continuous dcpos, which we call domain-complete spaces, and Gδ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces—in particular, all continuous dcpos—, all topologically complete spaces in the sense of Čech, and all quasi-Polish spaces—in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and ⊙-consonant—in particular, consonant; that the countably-based LCS-complete (resp., domain-complete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.

Almost Finiteness and the Small Boundary Property

Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and $m$-comparison for some nonnegative integer $m$ are all equivalent. The proof combines an Ornstein-Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz-Zhang and Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this implies that crossed product C$^*$-algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliott's program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property $\Gamma$, which under minimality confirms the Toms-Winter conjecture for such crossed products by the aforementioned work of Castillejos-Evington-Tikuisis-White-Winter.

Topological Spaces with the Freese–Nation Property

Abstract We give a proposal of generalization of the Freese–Nation property for topological spaces. We introduce a few properties related to Freese–Nation property: FNS, FN, FNS*, FNI. This article presents some relationship between these concepts. We show that spaces with the FNS property satisfy ccc and any product of such spaces also satisfies ccc. We show that all metrizable spaces have the FN property.

Common fixed point theorems in metric space using new CLR property

In this paper we present a new class of functions called Weakly CLR*_\( (f,g),T\) property which generalizes the class of mappings already defi ned in the literatures [8]. We obtain a result on Common fi xed point theorem in metric space for mappings satisfying weakly CLR*_\( (f,g),T\) property. Some examples to justify our results are given.