Let (Ω, d) be a metric space where Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension and let B be a partition of Ω. The coherent upper conditional prevision defined as the Choquet integral with respect to its associated Hausdorff outer measure is proven to satisfy the disintegration property and the conglomerative principle on every partition.

In the present paper, we give a new approach to Caristi's fixed pointtheorem on non-Archimedean fuzzy metric spaces. For this we define anordinary metric $d$ using the non-Archimedean fuzzy metric $M$ on a nonemptyset $X$ and we establish some relationship between $(X,d)$ and $(X,M,ast )$%. Hence, we prove our result by considering the original Caristi's fixedpoint theorem.

We show that a topological space is hereditarily irresolvable if and only if it is Hausdorff-reducible. We construct a compact irreducible T1-space and a connected Hausdorff space, each of which is strongly irresolvable. Furthermore, we show that the three notions of scattered, Hausdorff-reducible, and hereditarily irresolvable coincide for a large class of spaces, including metric, locally com...

The Hausdorff–Alexandroff Theorem states that any compact metric space is the continuous image of Cantor’s ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor’s ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here we present a compact counta...

The theory of Hausdorff dimension provides a general notion of the size of a set in a metric space. We define Hausdorff measure and dimension, enumerate some techniques for computing Hausdorff dimension, and provide applications to self-similar sets and Brownian motion. Our approach follows that of Stein [4] and Peres [3].

The aim of this paper is to study induced (quasi-)uniformities in Kramosil and Michalek's fuzzy metric spaces. Firstly, $I$-uniformity in the sense of J. Guti'{e}rrez  Garc'{i}a and $I$-neighborhood system in the sense of H"{o}hle and u{S}ostak are induced by the given fuzzy metric. It is shown that the fuzzy metric and the induced $I$-uniformity will generate the same $I$-neighborhood system. ...

We prove that each analytic set in R contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets. An essential part of the construction involves an analysis of Lipschitzlike mappings of separable metric s...

Kaneko and Sessa defined the concept of compatibility for multivalued mappings with Hausdorff metric and proved a coincidence point theorem. After then, Pathak defined the concept of weak compatibility and proved a coincidence theorem. In the present work, we define a new type compatibility for multivalued mappings with Hausdorff metric. This new type compatibility is different from compatibili...

We review basics concerning metric spaces from a contemporary viewpoint, and prove the important Baire category theorem, for both complete metric spaces and locally compact Hausdorff [1] spaces.