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| In [[mathematics]], a '''Priestley space''' is an [[partial order|ordered]] [[topological space]] with special properties. Priestley spaces are named after [[Hilary Priestley]] who introduced and investigated them.<ref>
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| Priestley, (1970).</ref> Priestley spaces play a fundamental role in the study of [[distributive lattice]]s. In particular, there is a [[duality theory for distributive lattices|duality]] between the [[category (mathematics)|category]] of Priestley spaces and the category of bounded distributive lattices.<ref>
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| Cornish, (1975).</ref>
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| <ref>Bezhanishvili et al. (2010)
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| </ref>
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| ==Definition==
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| A ''Priestley space'' is an ''ordered topological space'' {{math|(''X'',''τ'',≤)}}, i. e. a set {{math|''X''}} equipped with a [[partial order]] {{math|≤}} and a [[topological space#Definition|topology]] {{math|''τ''}}, satisfying
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| the following two conditions:
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| (i) {{math|(''X'',''τ'')}} is [[Compact space|compact]].
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| (ii) If <math>\scriptstyle x\,\not\le\, y</math>, then there exists a [[clopen set|clopen]] [[up-set]] {{math|''U''}} of {{math|''X''}} such that {{math|''x''<small>∈</small>''U''}} and {{math|''y''∉ ''U''}}. (This condition is known as the ''Priestley separation axiom''.)
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| ==Properties of Priestley spaces==
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| * Each Priestley space is [[Hausdorff space|Hausdorff]]. Indeed, given two points {{math|''x'',''y''}} of a Priestley space {{math|(''X'',''τ'',≤)}}, if {{math|''x''≠ ''y''}}, then as {{math|≤}} is a partial order, either <math>\scriptstyle x\,\not\le\, y</math> or <math>\scriptstyle y\,\not\le\, x</math>. Assuming, without loss of generality, that <math>\scriptstyle x\,\not\le\, y</math>, (ii) provides a clopen up-set {{math|''U''}} of {{math|''X''}} such that {{math|''x''<small>∈</small> ''U''}} and {{math|''y''∉ ''U''}}. Therefore, {{math|''U''}} and {{math|''V'' {{=}} ''X'' − ''U''}} are disjoint open subsets of {{math|''X''}} separating {{math|''x''}} and {{math|''y''}}.
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| * Each Priestley space is also [[zero-dimensional]]; that is, each [[open neighborhood]] {{math|''U''}} of a point {{math|''x''}} of a Priestley space {{math|(''X'',''τ'',≤)}} contains a clopen neighborhood {{math|''C''}} of {{math|''x''}}. To see this, one proceeds as follows. For each {{math|''y'' <small>∈</small> ''X'' − ''U''}}, either <math>\scriptstyle x\,\not\le\, y</math> or <math>\scriptstyle y\,\not\le\, x</math>. By the Priestley separation axiom, there exists a clopen up-set or a clopen [[down-set]] containing {{math|''x''}} and missing {{math|''y''}}. Obviously the intersection of these clopen neighborhoods of {{math|''x''}} does not meet {{math|''X'' − ''U''}}. Therefore, as {{math|''X''}} is compact, there exists a finite intersection of these clopen neighborhoods of {{math|''x''}} missing {{math|''X'' − ''U''}}. This finite intersection is the desired clopen neighborhood {{math|''C''}} of {{math|''x''}} contained in {{math|''U''}}.
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| It follows that for each Priestley space {{math|(''X'',''τ'',≤)}}, the topological space {{math|(''X'',''τ'')}} is a [[Stone space]]; that is, it is a compact Hausdorff zero-dimensional space.
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| Some further useful properties of Priestley spaces are listed below.
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| Let {{math|(''X'',''τ'',≤)}} be a Priestley space.
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| ::(a) For each closed subset {{math|''F''}} of {{math|''X''}}, both {{math|↑ ''F'' {{=}} {''x'' <small>∈</small> ''X'' : ''y'' ≤ ''x'' for some ''y'' <small>∈</small> ''F''} }} and {{math|↓ ''F'' {{=}} { ''x'' <small>∈</small> ''X'' : ''x'' ≤ ''y'' for some ''y'' <small>∈</small> ''F''} }} are closed subsets of {{math|''X''}}.
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| ::(b) Each open up-set of {{math|''X''}} is a union of clopen up-sets of {{math|''X''}} and each open down-set of {{math|''X''}} is a union of clopen down-sets of {{math|''X''}}.
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| ::(c) Each closed up-set of {{math|''X''}} is an intersection of clopen up-sets of {{math|''X''}} and each closed down-set of {{math|''X''}} is an intersection of clopen down-sets of {{math|''X''}}.
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| ::(d) Clopen up-sets and clopen down-sets of {{math|''X''}} form a [[subbasis]] for {{math|(''X'',''τ'')}}.
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| ::(e) For each pair of closed subsets {{math|''F''}} and {{math|''G''}} of {{math|''X''}}, if {{math|↑''F'' ∩ ↓''G'' {{=}} ∅}}, then there exists a clopen up-set {{math|''U''}} such that {{math|''F'' ⊆ ''U''}} and {{math|''U'' ∩ ''G'' {{=}} ∅}}.
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| A ''Priestley morphism'' from a Priestley space {{math|(''X'',''τ'',≤)}} to another Priestley space {{math|(''X''′,''τ''′,≤′)}} is a map {{math|f : ''X'' → ''X''′}} which is [[Continuous function (topology)|continuous]] and [[order-preserving]].
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| Let '''Pries''' denote the category of Priestley spaces and Priestley morphisms.
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| ==Connection with spectral spaces==
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| Priestley spaces are closely related to [[spectral space]]s. For a Priestley space {{math|(''X'',''τ'',≤)}}, let {{math|''τ''<sup>u</sup>}} denote the collection of all open up-sets of {{math|''X''}}. Similarly, let {{math|''τ''<sup>d</sup>}} denote the collection of all open down-sets of {{math|''X''}}.
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| Theorem:<ref>
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| Cornish, (1975). Bezhanishvili et al. (2010).</ref> If {{math|(''X'',''τ'',≤)}} is a Priestley space, then both {{math|(''X'',''τ''<sup>u</sup>)}} and {{math|(''X'',''τ''<sup>d</sup>)}} are spectral spaces.
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| Conversely, given a spectral space {{math|(''X'',''τ'')}}, let {{math|''τ''<sup>#</sup>}} denote the [[patch topology]] on {{math|''X''}}; that is, the topology generated by the subbasis consisting of compact open subsets of {{math|(''X'',''τ'')}} and their [[Complement (set theory)|complement]]s. Let also {{math|≤}} denote the [[specialization order]] of {{math|(''X'',''τ'')}}.
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| Theorem:<ref>
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| Cornish, (1975). Bezhanishvili et al. (2010).</ref> If {{math|(''X'',''τ'')}} is a spectral space, then {{math|(''X'',''τ''<sup>#</sup>,≤)}} is a Priestley space.
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| In fact, this correspondence between Priestley spaces and spectral spaces is [[functorial]] and yields an [[isomorphism]] between '''Pries''' and the category '''Spec''' of spectral spaces and [[spectral space|spectral map]]s.
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| ==Connection with bitopological spaces==
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| Priestley spaces are also closely related to [[bitopological space]]s.
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| Theorem:<ref>
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| Bezhanishvili et al. (2010).
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| </ref> If {{math|(''X'',''τ'',≤)}} is a Priestley space, then {{math|(''X'',''τ''<sup>u</sup>,''τ''<sup>d</sup>)}} is a [[pairwise Stone space]]. Conversely, if {{math|(''X'',''τ''<sub>1</sub>,''τ''<sub>2</sub>)}} is a pairwise Stone space, then {{math|(''X'',''τ'',≤)}} is a Priestley space, where {{math|''τ''}} is the join of {{math|''τ''<sub>1</sub>}} and {{math|''τ''<sub>2</sub>}} and {{math|≤}} is the specialization order of {{math|(''X'',''τ''<sub>1</sub>)}}.
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| The correspondence between Priestley spaces and pairwise Stone spaces is functorial and yields an isomorphism between the category '''Pries''' of Priestley spaces and Priestley morphisms and the category '''PStone''' of pairwise Stone spaces and [[bitopological space#Bi-continuity|bi-continuous map]]s. | |
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| Thus, one has the following isomorphisms of categories:
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| <center> <math>\mathbf{Spec}\cong \mathbf{Pries}\cong \mathbf{PStone}</math> </center>
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| One of the main consequences of the [[duality theory for distributive lattices]] is that each of these categories is dually equivalent to the category of bounded [[distributive lattice]]s.
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| ==See also==
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| * [[Spectral space]]
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| * [[Pairwise Stone space]]
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| * [[Distributive lattice]]
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| * [[Stone duality]]
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| * [[Duality theory for distributive lattices]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Priestley, H. A. (1970). Representation of distributive lattices by means of ordered Stone spaces.''Bull. London Math. Soc.'', (2) 186–190.
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| * Priestley, H. A. (1972). Ordered topological spaces and the representation of distributive lattices. ''Proc. London Math. Soc.'', 24(3) 507–530.
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| * Cornish, W. H. (1975). On H. Priestley's dual of the category of bounded distributive lattices. ''Mat. Vesnik'', 12(27) (4) 329–332.
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| * M. Hochster (1969). Prime ideal structure in commutative rings. ''Trans. Amer. Math. Soc.'', 142 43–60
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| * Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A. (2010). Bitopological duality for distributive lattices and Heyting algebras. ''Mathematical Structures in Computer Science'', 20.
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| [[Category:Topology]]
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| [[Category:Topological spaces]]
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Would you like to develop one mood and then make a surprising about-turn. A good introduction will interest the reader to read the full article. Proper research before writing an essay helps you to write an outstanding English essay. If you would like help in Research Papers and Term Paper Help you can visit Purchase Essay. quite well, he can become a free agent the subsequent season and create his own.
Also, they get quality assignments when they get aid from the companies. Amazingly company offers custom academic writing service to its valued customers worldwide in a thoroughly professional and dedicated manner. The company has the services of the best equipped editing team that has access to the best non-plagiarism software. If students need help with their dissertation format or term paper formats online writing services offer assistance in these aspects as well. Here's a checklist that you as a student can follow in order to make sure your essay is worth the effort:.
The thesis statement usually incorporates the main idea of the essay. Do they specify that the work needs to be double-spaced. Likewise to the similarities, the next part of writing a compare and contrast essay is to find out the differences among them. The tips on essay writing will help students write quality essays that will earn them high marks in the essay papers. This is important to ensure that you buy essay papers that are right, appropriate, and of high quality.
Particular requirements are really hard to provide for the reason that quite a bit relies on the particular writer’s experience with life, their own personal learning as well as on their particular expanding capability to criticise, gain knowledge along with a growing ability to appraise via what is learnt. There should also be an anecdote at the end of the essay in the concluding part. Get a pen, some paper, and paper printouts of your sources. This ensures that you know where the piece is going and do not deviate too far from the primary topic. The introduction should draw the intended audience attention, it should also build up the topic of discussion as you direct your reader into the essays argument.