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

A few characterizations of topological spaces with no infinite discrete subspace