On quantale-enriched categories the Hausdorff functor has no terminal coalgebra, but on enriched compact Hausdorff spaces it preserves codirected limits, making categories of Hausdorff polynomial coalgebras complete.
On the logic of generalised metric spaces
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Hausdorff coalgebras
On quantale-enriched categories the Hausdorff functor has no terminal coalgebra, but on enriched compact Hausdorff spaces it preserves codirected limits, making categories of Hausdorff polynomial coalgebras complete.