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.
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.
Reviewer: Bohumil František Šmarda (Brno)
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 |
References:
| [1] | 1 B. Banaschewski , The duality of distributive continuous lattices , Canadian J. of Math. XXXII ( 1980 ), 385 - 394 . MR 571932 | Zbl 0434.06011 · Zbl 0434.06011 · doi:10.4153/CJM-1980-030-3 |
| [2] | 2 B. Banaschewski and A. Pultr , Samuel compactification and completion of uniform frames , Math. Proc. Camb. Phil. Soc. 108 ( 1990 ), 63 - 78 . MR 1049760 | Zbl 0733.54020 · Zbl 0733.54020 · doi:10.1017/S030500410006895X |
| [3] | 3 K.H. Hofmann and J.D. Lawson , The spectral theory of distributive continuous lattices , Trans. Amer. Math. Soc. 246 ( 1978 ), 285 - 310 . MR 515540 | Zbl 0402.54043 · Zbl 0402.54043 · doi:10.2307/1997975 |
| [4] | 4 J.R. Isbell , Function spaces and adjoints , Math. Scand. 36 ( 1975 ), 317 - 339 . Article | MR 405340 | Zbl 0309.54016 · Zbl 0309.54016 |
| [5] | 5 P.T. Johnstone , ” Stone Spaces ”, Cambridge University Press , Cambridge , 1982 . MR 698074 | Zbl 0499.54001 · Zbl 0499.54001 |
| [6] | 6 P.T. Johnstone , Tychonoff’s theorem without the axiom of choice , Fund. Math. 113 ( 1981 ), 31 - 35 . Article | MR 641111 | Zbl 0503.54006 · Zbl 0503.54006 |
| [7] | 7 J.L. Kelley , The Tychonoff Product Theorem implies the Axiom of Choice , Fund. Math. 37 ( 1950 ), 75 - 76 . Article | MR 39982 | Zbl 0039.28202 · Zbl 0039.28202 |
| [8] | 8 H.A. Priestley , Representation of distributive lattices by means of ordered Stone spaces , Bull. London Math. Soc. 2 ( 1970 ), 186 - 190 . MR 265242 | Zbl 0201.01802 · Zbl 0201.01802 · doi:10.1112/blms/2.2.186 |
| [9] | 9 H.A. Priestley , Ordered topological spaces and the representation of distributive lattices , Proc. London Math. Soc. 24 ( 1972 ), 507 - 530 . MR 300949 | Zbl 0323.06011 · Zbl 0323.06011 · doi:10.1112/plms/s3-24.3.507 |
| [10] | 10 A. Pultr and J. Sichler , Frames in Priestley’s duality , Cahiers de Top. et Géom. Diff. Cat. XXIX - 3 ( 1988 ), 193 - 202 . Numdam | MR 975372 | Zbl 0666.54018 · Zbl 0666.54018 |
| [11] | 11 S. Vickers , ”Topology via Logic” , Cambrige Tracts in Theor. Comp. Sci. , Number 5 , Cambridge University Press , Cambridge , 1985 . MR 1002193 | Zbl 0668.54001 · Zbl 0668.54001 |
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