On ordinal-metric intersection topologies

A Classification of Ordinal Topologies

Ordinals carry a natural topology induced by their linear order. In this note, we classify the homeomorphism types of all ordinal topologies using the Cantor normal form and the notion of the Cantor–Bendixson rank of a point. Let < be a linear order on a set X. The order topology on X is generated by the subbase of rays of the form leftX(b) := {x ∈ X : x < b} and rightX(a) := {x ∈ X : a < x} fo...

Topologies for Probabilistic Metric Spaces

Topologies on Superspaces of TVS-Cone Metric Spaces

This paper investigates superspaces 𝒫0(X) and 𝒦0(X) of a tvs-cone metric space (X, d), where 𝒫0(X) and 𝒦0(X) are the space consisting of nonempty subsets of X and the space consisting of nonempty compact subsets of X, respectively. The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology (resp., the upper topology and the...

Exploring Intersection Trees for Indexing Metric Spaces

Searching in a dataset for objects that are similar to a given query object is a fundamental problem for several applications that use complex data. The general problem of many similarity measures for complex objects is their computational complexity, which makes them unusable for large databases. Here, we introduce a study of a variant of a metric tree data structure for indexing and querying ...

The Intersection of Topological and Metric Spaces

In this paper we will prove the Urysohn Meterization Theorem, which gives sufficient conditions for a topological space to be metrizable. In the process of proving this theorem, we will the discuss separation and countability axiom of topological spaces and prove the Urysohn Lemma.