A domain-theoretic Banach-Alaoglu theorem

We give a domain-theoretic analogue of the classical BanachAlaoglu theorem, showing that the patch topology on the weak* topology is compact. Various theorems follow concerning the stable compactness of spaces of valuations on a topological space. We conclude with reformulations of the patch topology in terms of polar sets or Minkowski functionals, showing, in particular, that the ‘sandwich set’ of linear functionals is compact.