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A Priestley view of spatialization of frames. (English) Zbl 0970.06006
The authors show how to apply the Priestley duality to obtain a straightforward characterization of spatial frames. In Priestley duality frames correspond to the LP-spaces (and LP-maps) which are described in the paper.
Main results: 1. An LP-space \(X\) is \(L\)-spatial iff for any two elements \(U\), \(V\) of the set of all decreasing clopen sets such that \(U\nsubseteq V\) there is an \(L\)-compact \(Y\subseteq X\) such that \(Y\subseteq U\) and \(Y\nsubseteq V\).
2. Let \(X\) be an \(L\)-compact \(L\)-regular LP-space. Let \(Y\) be a meet of a countable system of \(L\)-open subsets of \(X\). Then \(Y\) is \(L\)-spatial.
3. An LP-space is continuous iff it is locally compact.

MSC:
06D22 Frames, locales
06D05 Structure and representation theory of distributive lattices
06D10 Complete distributivity
54A05 Topological spaces and generalizations (closure spaces, etc.)
54D45 Local compactness, \(\sigma\)-compactness
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