Formal continuity implies uniform continuity near compact images on metric spaces

The localic completion of a metric space induces a canonical notion of continuous map between metric spaces. It is shown that these maps are continuous in the sense of Bishop constructive mathematics, i.e., uniformly continuous near every compact image.     

A continuity criterion for metric projections in banach spaces

The paper provides the complete characteristics of the class of reflexive strictly convex spaces in which the metric projections on each convex closed set are continuous.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 459–468, October, 1971.The author wishes to thank S. B. Stechkin for useful comments.     

Dimension of compact metric spaces

We give a survey of old and new results in dimension theory of compact metric spaces. Most of the relatively new results presented in the survey are based on the cohomological dimension approach. We complement the survey by stating the basics of cohomological dimension theory and listing some of its applications beyond the dimension theory.     

Covering compact metric spaces greedily

A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvátal’s analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of the n-dimensional sphere and n-dimensional Euclidean space to give short and transparent proofs of several best known bounds obtained from constructions in the literature on sphere coverings.     

Weak-open compact images of metric spaces

The main results of this paper are that (1) a space X is g-developable if and only if it is a weak-open π image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.     

Generic properties of compact metric spaces

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose elements enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.     

Universal uniform Eberlein compact spaces

A universal space is one that continuously maps onto all others of its own kind and weight. We investigate when a universal Uniform Eberlein compact space exists for weight . If , then they exist whereas otherwise, in many cases including , it is consistent that they do not exist. This answers (for many and consistently for all ) a question of Benyamini, Rudin and Wage of 1977.

     

Images of locally compact metric spaces

We establish relationships between locally compact-metric spaces and all kinds of spaces with mk-system (compact k-network) by means of compact-covering mappings, quotient mapping and closed mappings. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on the perturbations of compact quantum metric spaces

In this short note,we consider the perturbation of compact quantum metric spaces.We first show that for two compact quantum metric spaces(A,P) and(B,Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively,the quantum Gromov–Hausdorff distance between(A,P) and(B,Q) is small under certain conditions.Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.     

Expanding maps on compact metric spaces

The notions of locally expansive, positively expansive, expanding in the sense of Ruelle and expanding in the sense of Duvall and Husch are equivalent in a quite general setting.     

Embedding metric spaces in the rectilinear plane: A six-point criterion

We show that a metric space embeds in the rectilinear plane (i.e., isL ¹-embeddable in ℝ²) if and only if every subspace with five or six points does. A simple construction shows that for higher dimensionsk of the host rectilinear space the numberc(k) of points that need to be tested grows at least quadratically withk, thus disproving a conjecture of Seth and Jerome Malitz.     

Statistically localized sequences in metric spaces

In this paper we have introduced the statistically localized sequences in metric spaces and investigate basic properties of the statistically localized sequences. Also we have obtained some necessary and sufficient conditions for a localized sequence to be a statistically Cauchy sequence. It is also defined uniformly statistically localized sequences on metric spaces and its relation with statistically Cauchy sequences has been investigated.     

Open subspaces of locally compact metric spaces

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.     

Hyperspaces of compact sets in metric linear spaces

We write 2^x for the hyperspace of all non-empty compact sets in a complete metric linear space X topologized by the Hausdorff metric. Using the notation $\text{F}$(X) = {A ϵ 2^X: A is finite}, l^f₂ = {x} = (x_i) ϵ l₂: x_i = 0 for almost all i}, and l^σ₂ = {x = (x _i) ϵ l₂:σ^∞_i=1 (ix_i)² < ∞}, we have the following theorem:A family $\text{G}$ ⊂$\text{F}$(X) is homeomorphic to l^f₂ if $\text{G}$ is σ-fd-compact, the closure $\text{G}$ of $\text{G}$ in 2^x is not locally compact and if whenever A, B ∈ $\text{G}$, λ ∈ [0, 1] and C ⊂ λA + (1 - λ)B with card C⩽ max{card A, card B} then C ϵ $\text{G}$.Moreover, for any G_δ-AR-set $\text{G}$$\text{G}$ of $\text{G}$ with $\text{G}$$\text{G}$ ⊃ $\text{G}$ we have ($\text{G}$$\text{G}$, $\text{G}$)≅(l₂, l^ƒ₂).Similar conditions for hyperspaces to be homeomorphic to l^σ₂ are also established.     

A note on closed images of locally compact metric spaces

Summary A decomposition theorem about closed images of locally compact metric spaces is discussed. It is shown that a space is a closed image of a locally compact metric space if and only if it is a regular Fréchet space with a point-countable k-network, and each of its closed first-countable subset is locally compact.     

Positively expansive homeomorphisms on compact metric spaces

We give a new and simple proof to show that no homeomorphism of infinite compact metric spaces is positively expansive.     

On the uniform continuity of metric projections in Banach spaces

Let M be a convex Chebyshev subset of a uniformly convex and uniformly smooth Banach space. It is proved that the metric projection P_M of X onto M is uniformly continuous on every bounded subset of X. Moreover, a global and explicit estimate on the modulus of continuity of the metric projection is obtained.     

The potential method for conditional maxima in the locally compact metric spaces

Summary In the paper there is presented a method of reducing the constrained maximization problem to an unconditional one. The method consists of introducing a function, called potential, which at an unconditional maximum converges to the constrained maximum when a positive parameter of the potential tends to zero. A proof of such convergence for the locally compact metric spaces is given.