Complete Metrizability Research Articles

On the Borel complexity and the complete metrizability of spaces of metrics

Abstract Given a metrizable space X X , let A M ( X ) AM\left(X) be the space of continuous bounded admissible metrics on X X , which is endowed with the sup-metric. In this article, we shall investigate the Borel complexity and the complete metrizability of A M ( X ) AM\left(X) and show that a separable metrizable space X X is σ \sigma -compact if and only if A M ( X ) AM\left(X) is completely metrizable.

On certain new types of completeness properties using infinite chainability and associated metrization problems in uniform spaces

This article is in line with earlier investigations done in [10–12,14,15,18] and several such works. Here our aim is to introduce and study two new completeness-like properties, namely, Bourbaki quasi-completeness and cofinally Bourbaki quasi-completeness (we use infinite chains instead of finite ones), which strictly lie between compactness and completeness, primarily in the setting of uniform spaces. We use the concept of finite-component covers [19] to define a new type of modification of a uniform space, which plays a crucial role throughout the paper. In Section 2, we relate cBq-completeness to the existing notion of superparacompactness [19]. Another significant and very natural problem we deal with is the topological problem of metrizability of a uniform space using a Bq-complete and a cBq-complete metric. We obtain results similar to the classical Čech theorem about the complete metrizability of a metric space X in terms of its Stone-Čech compactification βX, which are presented in sections 3 and 4 of the article.

Fréchet subspaces of minimal usco and minimal cusco maps

Minimal usco/cusco maps are very important in functional analysis, optimization, selection theorems, the study of differentiability of Lipschitz functions, etc. We study topologies of uniform convergence on bornologies on the space of minimal usco and minimal cusco maps. We find sufficient conditions for metrizability and complete metrizability of minimal usco and minimal cusco maps equipped with the topology of uniform convergence on bornologies. We also investigate Fréchet locally convex subspaces of these spaces.

Topological Properties of the Space of Convex Minimal Usco Maps

We investigate the space of convex minimal usco maps from a Tychonoff space to the space of real numbers. Its elements are set-valued maps that are important e.g. in the study of subdifferentials of convex functions. We show that if the underlying space is normal, convex minimal usco maps can be approximated in the Vietoris topology by continuous functions. Using the strong Choquet game we prove complete metrizability of the space of convex minimal usco maps equipped with the upper Vietoris topology. We also study first countability, second countability and other properties of the (upper) Vietoris topology on this space.

Metrizability of the space of quasicontinuous functions

Let X be a topological space, (Y,d) be a metric space, Q(X,Y) be the space of quasicontinuous functions from X to Y and τUC be the topology of uniform convergence on compacta. We study first countability, metrizability and complete metrizability of (Q(X,Y),τUC). We will apply our results to characterize sequentially compact subsets of (Q(X,Y),τUC).

Completeness properties of the open-point and bi-point-open topologies on C(X)

This paper studies various completeness properties of the open-point and bi-point-open topologies on the space C(X) of all real-valued continuous functions on a Tychonoff space X. The properties range from complete metrizability to the Baire space property.

Preservation of complete metrizability by covering maps

A map f:X→Y is said to be w-covering if for every ordinal α and every compact S⊂Y such that the α-th Cantor derivative (S)α of S is a singleton {y} there is a compact subset E of X such that f(E)⊂S and (f(E))α={y}.We investigate the preservation of completely metrizability by various kinds of w-covering maps.

New types of completeness in metric spaces

This paper is devoted to introduce and study two new properties of completeness in the setting of metric spaces. We will call them Bourbaki-completeness and cofinal Bourbaki-completeness. These notions came from new classes of generalized Cauchy sequences appearing when we try to characterize the so-called Bourbaki-boundedness in a similar way that Cauchy sequences characterize the totally boundedness. We also study the topological problem of metrizability by means of a Bourbaki-complete or a cofinally Bourbaki-complete metric. At this respect, we obtain results in the line to the classical Cech theorem about the complete metrizability of a metric space X in terms of its Stone-Cech compactification beta X. Finally we give some relationships between these kinds of completeness and some properties related to paracompactness and uniform paracompactness in the framework of metrizable spaces.

Compact covers and function spaces

For a Tychonoff space X, we denote by Cp(X) and Cc(X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X in terms of Cp(X), we extend Okunevʼs results by showing that if there exists a surjection from Cp(X) onto Cp(Y) (resp. from Lp(X) onto Lp(Y)) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensenʼs theorem, we extend Pelantʼs result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from Cc(X) onto Cc(Y), then Y is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by ℓp-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and ℓp-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.

Note on reflexivity of some spaces of continuous integer-valued functions

Given a topological group G we denote by G∧ the group of characters on G and reflexivity of G means that the natural map from G to G∧∧ is a topological isomorphism.We show that for any zero-dimensional realcompact k-space X and a discrete finitely generated abelian group A, the group AX of continuous maps from X to A with pointwise addition and compact–open topology is reflexive, and we construct a countable non-reflexive closed subgroup of ZX, where X is a countable subspace of the plane (this group embeds as a closed subgroup in many copies of the product (∑Z)c of continuum of the discrete Specker group). We show also that, for metrizable separable X, analyticity of (ZX)∧ is equivalent to complete metrizability of X.

The metrizability and completeness of the σ-compact-open topology on [formula omitted

This paper studies the metrizability and various kinds of completeness properties of the space C σ ⁎ ( X ) of continuous real-valued bounded functions on a Tychonoff space X, where the function space has the σ-compact-open topology. The metrizability of C σ ⁎ ( X ) is studied in the context of generalized metric spaces and generalizations of first countability. On the other hand the completeness properties, studied here, range from complete metrizability to pseudocompleteness. Also the nuclear and related properties of C σ ⁎ ( X ) are studied.

Complete metrizability of topologies of strong uniform convergence on bornologies

We continue the study of topologies of strong uniform convergence on bornologies initiated by Beer and Levi (2009) [4]. In Beer and Levi (2009) [4] the metrizability of such topologies restricted to continuous functions was characterized and a compatible metric was displayed. We find necessary and sufficient conditions for completeness of the metric, for complete metrizability and for Polishness of topologies of strong uniform convergence on bornologies on continuous functions. Polishness of such topologies forces their coincidence with the compact-open topology.

Continuous selections and complete metrizability of the range

A survey of results showing that, in many selection theorems, the complete metrizability of the range is necessary as well as sufficient.

The metrizability and completeness of the support-open topology on [formula omitted

This paper studies the metrizability and various kinds of completeness property of the space C s ( X ) of continuous real-valued functions on a Tychonoff space X, where the function space has the support-open topology. The metrizability of C s ( X ) is studied in the context of generalized metric spaces and generalizations of the first countability. On the other hand the completeness properties, studied here, range from complete metrizability to pseudocompleteness.

Completeness properties of the pseudocompact-open topology on C(X)

Abstract This is a study of the completeness properties of the space C ps(X) of continuous real-valued functions on a Tychonoff space X, where the function space has the pseudocompact-open topology. The properties range from complete metrizability to the Baire space property.

Selectors in non-Archimedean spaces

Continuous selectors on the hyperspace F ( X ) are studied, when X is a non-Archimedean space. It is shown that a non-Archimedean space has a continuous selector if and only if it is topologically well orderable. Another characterization is given in terms of density and complete metrizability.

Complete metrizability of generalized compact-open topology

Let X and Y be Hausdorff topological spaces. Let P be the family of all partial maps from X to Y; a partial map is a pair ( B, f), where B ϵ CL( X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to Y. Denote by τ c the generalized compact-open topology on P . We show that if X is a hemicompact metrizable space and Y is a Fréchet space, then ( P, τ C ) is completely metrizable and homeomorphic to a closed subspace of ( CL( X), τ F ) × ( C( X, Y), τ CO ), where τ F is the Fell topology on CL( X) and τ CO is the compact-open topology on C( X, Y).

Completeness of Metrizable Pre-Images of Van Douwen-Complete Spaces

We shall show the recurrence of complete metrizability of irreducible closed pre-images of van Douwen-complete spaces. As its corollary we shall show that every van Douwen-complete space is ${G_\delta }$ in any Lašnev space.

Completeness of metrizable pre-images of van Douwen-complete spaces

We shall show the recurrence of complete metrizability of irreducible closed pre-images of van Douwen-complete spaces. As its corollary we shall show that every van Douwen-complete space is G δ {G_\delta } in any Lašnev space.

Classical set convergences and topologies

We provide some results, in a unified way, concerning the Hausdorff, Attoch-Wets, Vietoris, Fell, Wijsman topologies, defined on the closed subsets of a metric space and the Mosco topology defined on the closed convex subsets of a metric space. In particular we analyze countability axioms, metrizability and complete metrizability. Also, we give necessary and sufficient conditions for their pairwise coincidence.