We show that the canonical quantifications of uniform properties such as precompactness and total boundedness, which were already studied by Kuratowski and Hausdorff in the setting of complete metric spaces, can be generalized in the setting of products of metric spaces in an intuitively appealing way. 2000 Mathematics Subject Classification. 54E15, 18B30.

Based on the definitions and properties of fuzzy metric for random sets. We considered limit theory weighted sums sets in sense metric. The are independent compactly uniformly integrable, weights more general constants. convergence is induced by Hausdorff

This paper deals with three resolving parameters: the metric dimension, the upper dimension and the resolving number. We first answer a question raised by Chartrand and Zhang asking for a characterization of the graphswith equalmetric dimension and resolving number. We also solve in the affirmative a conjecture posed by Chartrand, Poisson and Zhang about the realization of the metric dimension ...

In this paper, following the ideas of (continuous) t-norm and interval numbers, a concept of (continuous) interval-valued t-norm is proposed. Based on the interval-valued fuzzy set and the continuous interval-valued t-norm, we propose a notion of interval-valued fuzzy metric space, which is a generalization of fuzzy metric space in the sense of George and Veeramani [Fuzzy Sets and Systems 64 (1...

Considering the increasing interest in fuzzy theory and possible applications, the concept of fuzzy metric space concept has been introduced by several authors from different perspectives. This paper interprets the theory in terms of metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology. We study interrelationships between this theory and other fuzzy theories such as intuit...

The geometric structural complexity of spatial objects does not render an intuitive distance metric on the data space that measures spatial proximity. However, such a metric provides a formal basis for analytical work in transformation-based multidimensional spatial access methods, including locality preservation of the underlying transformation and distance-based spatial queries. We study the ...

A natural approach to defining continuous change of shape is in terms of a metric that measures the difference between two regions. We consider four such metrics over regions: the Hausdorff distance, the dual-Hausdorff distance, the area of the symmetric difference, and the optimal-homeomorphism metric. Each of these gives a different criterion for continuous change. We establish qualitative pr...

We give an alternative proof of Fedorchuk’s recent result that dimX6DgX for compact Hausdorff spaces X. We use the Löwenheim-Skolem theorem to reduce the problem to the metric case.

We characterize finite-dimensional normed linear spaces as strongly proximinal subspaces in all their superspaces. A connection between upper Hausdorff semi-continuity of metric projection and finite dimensionality of subspace is given.

In this paper we announce various new results about singularities in the mean curvature flow. Some results apply to any weak solution (i.e., any Brakke flow of integral varifolds.) Our strongest results, however, are for initially regular mean-convex hypersurfaces. (We say a hypersurface is mean-convex if it bounds a region such that the mean curvature with respect to the inward unit normal is ...