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We give multiple descriptions of a topological universe of finitary sets,which can be seen as a natural limit completion of the hereditarily finitesets. This universe is characterized as a metric completion of the hereditarilyfinite sets; as a Stone space arising as the solution of a functorialfixed-point equation involving the Vietoris construction; as the Stone dual ofthe free modal algebra; and as the subspace of maximal elements of a domainequation involving the Plotkin (or convex) powerdomain. These resultsillustrate the methods developed in the author's 'Domain theory in logicalform', and related literature, and have been taken up in recent work ontopological coalgebras.  The set-theoretic universe of finitary sets also supports an interesting formof set theory. It contains non-well founded sets and a universal set; and isclosed under positive versions of the usual axioms of set theory.

A Cook's Tour of the Finitary Non-Well-Founded Sets