An Adaptation of the Vietoris Topology for Ordered Compact Sets
Christopher Caruvana, Jared Holshouser
TL;DR
This paper develops the Vietoris power topology $\mathsf V(X^\kappa)$ as an ordered analogue of the Vietoris hyperspace to study ordered compact subsets via a product-like topology. It situates $\mathsf V(X^\kappa)$ relative to classical product topologies and analyzes covering properties through selection principles, showing that such properties do not, in general, transfer from the ground space to the Vietoris power. Key findings include that $\mathsf V(X^\kappa)$ can fail to be compact even for compact $X$, and can fail Lindelöf or Menger in $\mathsf V(\mathbb{R}^\omega)$, with detailed propeties characterized in discrete and countable settings (e.g., $\mathbb K(\omega,\mathrm{ord})$). The paper also proves that the image map $f \mapsto \mathrm{img}(f)$ from ordered to unordered compact sets is continuous and, for infinite $\kappa$, open onto its range, clarifying how ordered/unordered comparisons behave within this new framework and enabling construction of new spaces with preserved density properties.
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 Lindelöf, 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.