Point Networks for Special Subspaces of $\mathbb{R}^{\kappa}$

TL;DR

This paper provides uniform characterizations of special subspaces of product spaces of lines using collections of subsets organized by directed sets, leading to new characterizations of Eberlein, Talagrand, and Gulko compacta.

Contribution

It introduces novel uniform characterizations of certain subspaces of product spaces, connecting them with bases, networks, and point networks organized by directed sets.

Findings

New characterizations of Eberlein compacta

Characterizations of Talagrand compacta

Characterizations of Gulko compacta

Abstract

Uniform characterizations of certain special subspaces of products of lines are presented. The characterizations all involve a collection of subsets (base, almost subbase, network or point network) organized by a directed set. New characterizations of Eberlein, Talagrand and Gulko compacta follow.