Descriptive Proximities I: Properties and interplay between classical proximities and overlap

Abstract

The theory of descriptive nearness is usually adopted when dealing with sets that share some common properties even when the sets are not spatially close, i.e., the sets have no members in common. Set description results from the use of probe functions to define feature vectors that describe a set and the nearness of sets is given by their proximities. A probe on a non-empty set $X$ is a real-valued function $\Phi: X \rightarrow \mathbb{R}^n$, where $\Phi(x)= (\phi_1(x),.., \phi_n(x))$. We establish a connection between relations on an object space $X$ and relations on the feature space $\Phi(X).$ Having as starting point the Peters proximity, two sets are \emph{descriptively near}, if and only if their descriptions intersect. In this paper, we construct a theoretical approach to a more visual form of proximity, namely, descriptive proximity, which has a broad spectrum of applications. We organize descriptive proximities on two different levels: weaker or stronger than the Peters proximity. We analyze the properties and interplay between descriptions on one side and classical proximities and overlap relations on the other side.