Coproducts of proximity spaces

TL;DR

This paper introduces coproducts of proximity spaces, explores their properties, and analyzes the Smirnov compactification of these coproducts, revealing their topological and dimensional characteristics, including conditions for metrizability and homeomorphism to Stone-Cech compactifications.

Contribution

It defines coproducts of proximity spaces and studies their Smirnov compactifications, providing new insights into their topological properties and relationships with classical compactifications.

Findings

Coproducts of proximity spaces can be densely embedded in the Smirnov compactification of their coproducts.

The Smirnov compactification of an infinite coproduct is never metrizable.

The proximity dimension of the coproduct's Smirnov compactification equals the supremum of the individual dimensions.

Abstract

In this paper, we introduce coproducts of proximity spaces. After exploring several of their basic properties, we show that given a collection of proximity spaces, the coproduct of their Smirnov compactifications proximally and densely embeds in the Smirnov compactification of the coproduct of the original proximity spaces. We also show that the dense proximity embedding is a proximity isomorphism if and only if the index set is finite. After constructing a number of examples of coproducts and their Smirnov compactifications, we explore several properties of the Smirnov compactification of the coproduct, including its metrizability, connectedness of the boundary, dimension, and its relation to the Stone-Cech compactification. In particular, we show that the Smirnov compactification of the infinite coproduct is never metrizable and that its boundary is highly disconnected. We also show that the proximity dimension of the Smirnov compactification of the coproduct equals the supremum of the covering dimensions of the individual Smirnov compactifications and that the Smirnov compactification of the coproduct is homeomorphic to the Stone-Cech compactification if and only if each individual proximity space is equipped with the Stone-Cech proximity. We finish with an example of a coproduct with the covering dimension $0$ but the proximity dimension $\infty.$