An Adaptation of the Vietoris Topology for Ordered Compact Sets

TL;DR

This paper introduces a new topology inspired by the Vietoris topology for powers of a space, compares it with other product topologies, and explores its properties, especially for discrete and Euclidean spaces.

Contribution

It adapts the Vietoris topology for ordered compact sets and analyzes its topological properties in different contexts, highlighting differences from classical Vietoris topology.

Findings

The topology behaves differently on discrete spaces, with specific properties identified.

On Euclidean spaces, the power is not Lindelöf or Menger, unlike the classical Vietoris topology.

Covering properties of the ground space do not necessarily transfer to the Vietoris power.

Abstract

We discuss a natural topology on powers of a space that is inspired by the Vietoris topology on compact subsets. We then place this topology in context with other product topologies; specifically, we compare this topology with the Tychonoff product, the box product, and Bell's uniform box topology. We identify a variety of topological properties for the specific case when the ground space is discrete. When the ground space is the Euclidean real line, we show that the resulting power is not Lindel\"{o}f, and hence, not Menger. This shows that, unlike the the Vietoris topology on unordered compact subsets, covering properties of the ground space need not transfer to the Vietoris power.