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...

We prove that the extrinsic Hausdorff dimension is always greater than or equal to the intrinsic Hausdorff dimension in models of triangulated random surfaces with action which is quadratic in the separation of vertices. We furthermore derive a few naive scaling relations which relate the intrinsic Hausdorff dimension to other critical exponents. These relations suggest that the intrinsic Hausd...

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension n is a Hausdorff topological space which has a countable base of...

We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conque...

We call a Gromov-Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. In this paper, we prove that any Ricci limit space has integral Hausdorff dimension provided that its Hausdorff dimension is not greater than two. We also classify one-dimensional Ricci limit spaces.

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 this paper, two cardinal inequalities for functionally Hausdorff spaces are established. A bound on the cardinality of the τθ-closed hull of a subset of a functionally Hausdorff space is given. Moreover, the following theorem is proved: ifX is a functionally Hausdorff space, then |X| ≤ 2χ(X)wcd(X).

A flow of an open manifold is very complicated even if its orbit space is Hausdorff. In this paper, we define the strongly Hausdorff flows and consider their dynamical properties in terms of the orbit spaces. By making use of this characterization, we finally classify all the strongly Hausdorff C1-flows.

We prove that the topology of a compact Hausdorff topological Heyting algebra is a Stone topology. It then follows from known results that a Heyting algebra is profinite iff it admits a compact Hausdorff topology that makes it a compact Hausdorff topological Heyting algebra.