  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.
