A few characterizations of topological spaces with no infinite discrete subspace

TL;DR

This paper characterizes FAC topological spaces, which lack infinite discrete subspaces, through properties like finite unions of irreducible closed sets and dense Noetherian subspaces, extending classical poset results.

Contribution

It provides new characterizations of FAC spaces, connecting them to properties like dense Noetherian subspaces and bounds on relatively Hausdorff subsets, extending known poset theorems.

Findings

Every closed set is a finite union of irreducible closed subsets.

FAC spaces are characterized by the presence of dense Noetherian subspaces.

In FAC spaces, the sizes of relatively Hausdorff subsets are bounded.

Abstract

We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible closed subsets. The full result extends well-known characterizations of posets with no infinite antichain. One of them is that FAC spaces are, equivalently, topological spaces in which every closed set contains a dense Noetherian subspace, or spaces in which every Hausdorff subspace is finite, or in which no subspace has any infinite relatively Hausdorff subset. The latter comes with a nice min-max property, extending an observation of Erd\"os and Tarski in the case of posets: on spaces with no infinite relatively Hausdorff subset, the cardinalities of relatively Hausdorff subsets are bounded, and the least upper bound is also the least cardinality of a family of closed irreducible subsets that cover the space.