A domain-theoretic Banach–Alaoglu theorem

Authors:
Gordon Plotkin
Affiliations:
School of Informatics, University of Edinburgh, King's Buildings, Edinburgh EH9 3JZ, U.K.
Venue:
Mathematical Structures in Computer Science
Year:
2006

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Abstract

We give a domain-theoretic analogue of the classical Banach–Alaoglu 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.