Using the Interval Metric for Modeling Entities Geometrics in R2 – Case Study Interval Circumference

The study of some distances provide science away to separate two entities.It has applications in various fields such as remote sensing, datamining, pattern recognition and multivariate data analysis and others. If the distance is a Hausdorff metric, the guarantee is that all individuals are available. With the use of the distance of Trindade et al, we intend to extend the real topology to an interval topology, since the interval distance preserves the uncertainties and exits noise in the data. The present work proposes an interval circumference using an interval distance of a point to the center (pixel), like a set of pixels obeying certain distances to the center. With the interval circumference we intend to extend the notion of open ball and the concepts of neighborhood for the construction of the interval topology. A circumference separates a space into three regions, inner region, border region and outer region, where we construct our notion of neighborhood. In this work we will explore only the geometric properties of the intervalcircumference, we will extrapolate the notion from point to pixel by providing a differentiated frontier region for the clustering area.

On Completeness of Sliced Spaces under the Alexandrov Topology

We show that in a sliced spacetime ( V , g ) , global hyperbolicity in V is equivalent to T A -completeness of a slice, if and only if the product topology T P , on V, is equivalent to T A , where T A denotes the usual spacetime Alexandrov “interval” topology.

Split Hausdorff internal topologies on posets

In this paper, the concepts of weak quasi-hypercontinuous posets and weak generalized finitely regular relations are introduced. The main results are: (1) when a binary relation ρ : X ⇀ Y satisfies a certain condition, ρ is weak generalized finitely regular if and only if (φρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φρ(X, Y), ⊆) is split T2; (2) the relation ≰ on a poset P is weak generalized finitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T2.

Interval topology in contact geometry

A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.