7209:
7171:
7164:
7079:
7072:
7065:
6990:
6983:
6976:
6963:
6956:
6949:
6895:
6888:
6881:
6868:
6861:
6783:
6740:
6733:
6712:
6705:
6698:
6566:
6559:
6495:
6474:
6420:
6413:
6399:
6345:
6338:
6331:
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6277:
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6140:
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5923:
5916:
5909:
5902:
5895:
5888:
5843:
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5822:
5815:
5808:
5778:
5771:
5764:
5757:
5750:
5743:
5736:
5729:
7178:
7086:
7223:
7216:
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7143:
7124:
7117:
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5722:
5715:
3512:
the open complexes generate all the complexes as a
Boolean algebra. These related representations provide a well defined mathematical apparatus for studying the relationship between truth modalities (possibly true vs necessarily true, studied in modal logic) and notions of provability and refutability (studied in intuitionistic logic) and is thus deeply connected to the theory of
4845:
3527:
sets trivially form a topological field of sets as each clopen set is its own interior and closure. The Stone representation of a
Boolean algebra can be regarded as such a topological field of sets, however in general the topology of a topological field of sets can differ from the topology generated
3506:
Every interior algebra can be represented as a topological field of sets with the underlying
Boolean algebra of the interior algebra corresponding to the complexes of the topological field of sets and the interior and closure operators of the interior algebra corresponding to those of the topology.
3511:
can be represented by a topological field of sets with the underlying lattice of the
Heyting algebra corresponding to the lattice of complexes of the topological field of sets that are open in the topology. Moreover the topological field of sets representing a Heyting algebra may be chosen so that
3551:
and provide a generalization of the Stone representation of
Boolean algebras. Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the
2530:
onto the two element
Boolean algebra and form complexes by associating each element of the Boolean algebra with the set of such homomorphisms that map it to the top element. (The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the top elements under
2498:
and that an atom is below an element of a finite
Boolean algebra if and only if that element is contained in the ultrafilter corresponding to the atom. This leads us to construct a representation of a Boolean algebra by taking its set of ultrafilters and forming complexes by associating with each
987:
865:
2493:
In the case of
Boolean algebras which are not complete and atomic we can still generalize the power set representation by considering fields of sets instead of whole power sets. To do this we first observe that the atoms of a finite Boolean algebra correspond to its
3694:
Like the topological fields of sets, preorder fields play an important role in the representation theory of interior algebras. Every interior algebra can be represented as a preorder field with its interior and closure operators corresponding to those of the
4647:
2591:
generated by the complexes of a field of sets. (It is just one of notable topologies on the given set of points; it often happens that another topology is given, with quite different properties, in particular, not zero-dimensional). Given a field of sets
4455:
3564:
after the mathematicians who first generalized Stone's result for
Boolean algebras to interior algebras and should not be confused with the Stone topology of the underlying Boolean algebra of the interior algebra which will be a finer topology).
3813:
3896:
875:
753:
1507:
1353:
3912:, the preorder represents the accessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worlds in which individual sentences in the theory hold, providing a representation of the
4095:
of the topology of the Stone representation.) While representation of modal algebras by general modal frames is possible for any normal modal algebra, it is only in the case of interior algebras (which correspond to the modal logic
499:
743:
672:
4348:
3543:
If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology.
3556:
of the interior algebra (which form a base for a topology). These complexes are then precisely the open complexes and the construction produces a Stone field representing the interior algebra - the
2499:
element of the
Boolean algebra the set of ultrafilters containing that element. This construction does indeed produce a representation of the Boolean algebra as a field of sets and is known as the
1566:
1412:
4643:
7022:
6927:
6829:
4353:
3297:
2077:
7697:
2634:
561:
5689:
4172:
4934:
5623:
5574:
5656:
3732:
3818:
3617:
7640:
3112:
7260:
5347:
596:
4014:
3727:
3472:
3218:
simply for the underlying set of a probability space, particularly in the case where every subset is an event.) Measure spaces and probability spaces play a foundational role in
2359:
2107:
1422:
1268:
1074:
7551:
5525:
4268:
3689:
3373:
3333:
3148:
2705:
1803:
1215:
309:
83:
7582:
7516:
4224:
2942:
2859:
2800:
2763:
2669:
7664:
7289:
6173:
6042:
6010:
5376:
5266:
4879:
3958:
3429:
3401:
2973:
2887:
2729:
1962:
1827:
1243:
1146:
1098:
1016:
359:
133:
4840:{\displaystyle f_{i}(S_{1},\ldots ,S_{n})=\left\{x\in X:{\text{ there exist }}x_{1}\in S_{1},\ldots ,x_{n}\in S_{n}{\text{ such that }}R_{i}(x_{1},\ldots ,x_{n},x)\right\}}
7485:
7426:
6127:
5496:
5408:
2995:
2909:
2826:
4072:
theories are always algebraic, the complexes determining the preorder being the sets of possible worlds in which the sentences of the theory closed under necessity hold.
447:
2531:
these homomorphisms.) With this approach one sees that Stone representation can also be regarded as a generalization of the representation of finite Boolean algebras by
2166:
2140:
3649:
1886:
1673:
1938:
685:
614:
7400:
7356:
5467:
5438:
5288:
4484:
3984:
4066:
4040:
2329:
2303:
2277:
2196:
7201:
7109:
4567:
4511:
1989:
1767:
1740:
1634:
4540:
3192:
3168:
1604:
1177:
411:
386:
211:
4899:
4587:
2581:
2466:
2442:
2422:
2399:
2379:
2242:
2218:
1906:
1847:
1713:
1693:
1118:
1044:
526:
441:
332:
254:
179:
157:
106:
5116:
2504:
4083:(specialization preorder) we obtain a representation of the interior algebra as a canonical preorder field. By replacing the preorder by its corresponding
3009:
of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexes of the Stone representation. The area of mathematics known as
982:{\displaystyle F_{1}\cap \cdots \cap F_{n}\in {\mathcal {F}}{\text{ for all integers }}n\geq 1{\text{ and all }}F_{1},\ldots ,F_{n}\in {\mathcal {F}}.}
860:{\displaystyle F_{1}\cup \cdots \cup F_{n}\in {\mathcal {F}}{\text{ for all integers }}n\geq 1{\text{ and all }}F_{1},\ldots ,F_{n}\in {\mathcal {F}}.}
3241:
which already have a topology associated with them - this should not be confused with the topology generated by taking arbitrary unions of complexes.
3528:
by taking arbitrary unions of complexes and in general the complexes of a topological field of sets need not be open or closed in the topology.
3013:
is founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a
4281:
5308:
4947:
Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is
3481:
4108:
The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbitrary (normal)
4092:
3005:
The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space is known as the
1512:
1358:
4592:
6997:
6902:
6804:
2479:; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This
4087:
we obtain an alternative representation of the interior algebra as a topological field of sets. (The topology of this "
3900:
Similarly to topological fields of sets, preorder fields arise naturally in modal logic where the points represent the
3256:
1998:
7669:
4084:
5184:
2595:
533:
5664:
4115:
4904:
5582:
5533:
5137:
The listed statements are equivalent if (1) and (2) hold. The equivalence of statements (a) and (b) follows from
3920:
which are fields of sets with an additional accessibility relation providing representations of modal algebras.
5631:
5301:
5030:
4109:
3499:
respectively. Topological fields of sets representing these algebraic structures provide a related topological
3580:
7587:
5225:
3913:
3075:
3553:
7433:
7238:
5695:
5325:
3233:
we often deal with measure spaces and probability spaces derived from rich mathematical structures such as
4450:{\displaystyle {\mathcal {C}}(\mathbf {X} )=({\mathcal {F}},\cap ,\cup ,\prime ,\emptyset ,X,(f_{i})_{I})}
4079:. Given an interior algebra, by replacing the topology of its Stone representation with the corresponding
568:
5220:
3989:
3702:
3476:
Topological fields of sets play a fundamental role in the representation theory of interior algebras and
3438:
2472:
2334:
2082:
1049:
1023:
17:
7530:
5504:
4237:
3658:
3342:
3302:
3117:
2674:
2512:
1772:
1184:
278:
52:
3061:
theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called
2552:) if and only if for every pair of distinct points there is a complex containing one and not the other.
7720:
7560:
7494:
5294:
5215:
4177:
4080:
3038:
2916:
2833:
2774:
2737:
2643:
678:
218:
7645:
7270:
6148:
6017:
5985:
5357:
5247:
4860:
3939:
3410:
3382:
2954:
2868:
2710:
1943:
1808:
1224:
1127:
1079:
997:
340:
114:
7715:
3214:
and represent properties of outcomes for which we wish to assign probabilities. (Many use the term
2495:
2484:
7464:
7411:
6106:
5475:
5392:
3808:{\displaystyle \mathrm {Int} (S)=\{x\in X:{\text{ there exists a }}y\in S{\text{ with }}y\leq x\}}
2978:
2892:
2809:
4854:
3891:{\displaystyle \mathrm {Cl} (S)=\{x\in X:{\text{ there exists a }}y\in S{\text{ with }}x\leq y\}}
420:
187:
2145:
2112:
5232:
4231:
3622:
2766:
1852:
1639:
46:
1911:
7385:
7341:
7291:
5446:
5417:
5378:
5273:
4460:
3963:
3014:
2487:
2476:
264:
4045:
4019:
2308:
2282:
2247:
2171:
1502:{\displaystyle \bigcap _{i=1}^{\infty }F_{i}:=F_{1}\cap F_{2}\cap \cdots \in {\mathcal {F}}}
1348:{\displaystyle \bigcup _{i=1}^{\infty }F_{i}:=F_{1}\cup F_{2}\cup \cdots \in {\mathcal {F}}}
7186:
7134:
7094:
7035:
4545:
4489:
4227:
3496:
1967:
1745:
1718:
1612:
257:
4516:
3177:
3153:
8:
6747:
5138:
4980:
4959:
4850:
3696:
3234:
2560:
2516:
2221:
225:
4100:) that the general modal frame corresponds to topological field of sets in this manner.
1586:
1159:
393:
368:
193:
7522:
5861:
5172:
5074:
4884:
4572:
3517:
3500:
3223:
3195:
3054:
3031:
2566:
2451:
2427:
2407:
2384:
2364:
2227:
2203:
1891:
1832:
1698:
1678:
1103:
1029:
607:
511:
426:
317:
239:
214:
164:
142:
91:
2168:). One notable consequence: the number of complexes, if finite, is always of the form
7299:
5180:
3524:
3404:
3336:
3238:
2588:
312:
86:
7449:
5786:
494:{\displaystyle X\setminus F\in {\mathcal {F}}{\text{ for all }}F\in {\mathcal {F}}.}
5154:
5065:
5018:
3905:
3432:
3376:
3407:
i.e. the closure and interior of every complex is also a complex. In other words,
3210:
and represent potential outcomes while the measurable sets (complexes) are called
1573:
267:
of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
4989:
3547:
Topological fields of sets that are separative, compact and algebraic are called
3513:
3508:
3477:
2803:
2637:
738:{\displaystyle F\cap G\in {\mathcal {F}}{\text{ for all }}F,G\in {\mathcal {F}}.}
667:{\displaystyle F\cup G\in {\mathcal {F}}{\text{ for all }}F,G\in {\mathcal {F}}.}
233:
28:
7554:
7488:
6940:
5940:
5243:
5044:
5035:
4103:
3219:
3171:
3058:
2538:
335:
109:
7709:
6178:
6069:
5202:
Handbook of Modal Logic, Volume 3 of Studies in Logic and Practical Reasoning
5110:
5024:
5013:
4983: – topology in which the intersection of any family of open sets is open
3917:
3042:
3010:
2945:
2862:
1992:
1260:
27:"Set algebra" redirects here. For the basic properties and laws of sets, see
4857:
and relations as operators can be viewed as a special case of relations. If
256:" is used in the sense of a Boolean algebra and should not be confused with
6284:
6202:
5386:
5086:
5080:
4995:
4963:
2583:
the intersection of all the complexes contained in the filter is non-empty.
2527:
2520:
2508:
2445:
34:
Algebraic concept in measure theory, also referred to as an algebra of sets
6590:
5161:, Logic Journal of the IGPL, Volume 8, Issue 4, p. 393-450, July 2000
5092:
1247:
5098:
4948:
3540:
if and only if there is a base for its topology consisting of complexes.
3485:
3006:
2532:
229:
38:
6668:
6437:
6362:
5197:, Ph.D. thesis, University of Cape Town Department of Mathematics, 1991
5104:
5004:
2997:
is both separative and compact (in which case it is described as being
2949:
1019:
5119: – Every Boolean algebra is isomorphic to a certain field of sets
4343:{\displaystyle \mathbf {X} =(X,\left(R_{i}\right)_{I},{\mathcal {F}})}
7455:
6842:
5704:
5056:
5027: – Family closed under complements and countable disjoint unions
3492:
3035:
2731:
is the topology formed by taking arbitrary unions of complexes. Then
1607:
505:
183:
4849:
This construction can be generalized to fields of sets on arbitrary
3652:
5141:. This is also true of the equivalence of statements (c) and (d).
4350:
on a relational structure, is the Boolean algebra with operators
3560:. (The topology of the Stone representation is also known as the
3230:
1769:-element). It appears that every finite field of sets (it means,
2475:
can be represented as a power set – the power set of its set of
5200:
Patrick Blackburn, Johan F.A.K. van Benthem, Frank Wolter ed.,
4958:
was first used in the case where the algebraic structure was a
1574:
Fields of sets in the representation theory of Boolean algebras
362:
136:
7177:
7085:
565:
Assuming that (1) holds, this condition (2) is equivalent to:
5168:, Annals of Pure and Applied Logic, 44, p. 173-242, 1989
4104:
Complex algebras and fields of sets on relational structures
4075:
A separative compact algebraic preorder field is said to be
3531:
2539:
Separative and compact fields of sets: towards Stone duality
2361:
that is an atom; the latter means that a nonempty subset of
186:
as an element, and is closed under the operations of taking
5083: – Family closed under unions and relative complements
3020:
3923:
1251:
if the following additional condition (4) is satisfied:
5101: – Family closed under subsets and countable unions
3480:. These two classes of algebraic structures provide the
5179:(3rd ed.). Cambridge: Cambridge University Press.
5070:
Pages displaying short descriptions of redirect targets
5040:
Pages displaying short descriptions of redirect targets
5038: – Generalization of mass, length, area and volume
3960:
which determines the preorder in the following manner:
4403:
1675:
of this set and its power set) is a field of sets. If
1561:{\displaystyle F_{1},F_{2},\ldots \in {\mathcal {F}}.}
1407:{\displaystyle F_{1},F_{2},\ldots \in {\mathcal {F}}.}
1255:
Any/both of the following equivalent conditions hold:
7672:
7648:
7590:
7563:
7533:
7497:
7467:
7414:
7388:
7344:
7273:
7241:
7189:
7097:
7000:
6905:
6807:
6151:
6109:
6020:
5988:
5667:
5634:
5585:
5536:
5507:
5478:
5449:
5420:
5395:
5360:
5328:
5276:
5250:
4907:
4887:
4863:
4650:
4638:{\displaystyle S_{1},\ldots ,S_{n}\in {\mathcal {F}}}
4595:
4575:
4548:
4519:
4492:
4463:
4356:
4284:
4240:
4180:
4118:
4048:
4022:
3992:
3966:
3942:
3821:
3735:
3705:
3661:
3625:
3583:
3441:
3413:
3385:
3345:
3305:
3259:
3180:
3156:
3120:
3078:
2981:
2957:
2919:
2895:
2871:
2836:
2812:
2777:
2740:
2713:
2677:
2646:
2598:
2569:
2454:
2430:
2410:
2387:
2367:
2337:
2311:
2285:
2250:
2230:
2206:
2174:
2148:
2115:
2085:
2001:
1970:
1946:
1940:
that establishes a one-to-one correspondence between
1914:
1894:
1855:
1849:
may be infinite) admits a representation of the form
1835:
1811:
1775:
1748:
1721:
1701:
1681:
1642:
1615:
1589:
1515:
1425:
1361:
1271:
1227:
1187:
1162:
1130:
1106:
1082:
1052:
1032:
1000:
878:
756:
688:
617:
602:
Any/all of the following equivalent conditions hold:
571:
536:
514:
450:
429:
396:
371:
343:
320:
281:
242:
196:
167:
145:
117:
94:
55:
5107: – Family of sets closed under countable unions
5077: – Branch of mathematics concerning probability
5049:
Pages displaying wikidata descriptions as a fallback
5000:
Pages displaying wikidata descriptions as a fallback
4985:
Pages displaying wikidata descriptions as a fallback
3025:
3017:
exists between Boolean algebras and Boolean spaces.
7017:{\displaystyle \varnothing \not \in {\mathcal {F}}}
6922:{\displaystyle \varnothing \not \in {\mathcal {F}}}
6824:{\displaystyle \varnothing \not \in {\mathcal {F}}}
5117:
Stone's representation theorem for Boolean algebras
4992: – Identities and relationships involving sets
4951:to the complex algebra corresponding to the field.
3030:If an algebra over a set is closed under countable
2505:
Stone's representation theorem for Boolean algebras
7691:
7658:
7634:
7576:
7545:
7510:
7479:
7420:
7394:
7350:
7283:
7254:
7195:
7103:
7016:
6921:
6823:
6167:
6121:
6036:
6004:
5683:
5650:
5617:
5568:
5519:
5490:
5461:
5432:
5402:
5370:
5341:
5282:
5260:
5068: – Algebraic object with an ordered structure
5052:Pages displaying short descriptions with no spaces
4928:
4893:
4873:
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4505:
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4342:
4262:
4218:
4166:
4060:
4034:
4008:
3978:
3952:
3890:
3807:
3721:
3683:
3643:
3611:
3466:
3423:
3395:
3367:
3327:
3291:
3186:
3162:
3142:
3106:
2989:
2967:
2936:
2903:
2881:
2853:
2820:
2794:
2757:
2723:
2699:
2663:
2628:
2575:
2460:
2436:
2416:
2393:
2373:
2353:
2323:
2297:
2271:
2236:
2212:
2190:
2160:
2134:
2101:
2071:
1983:
1956:
1932:
1900:
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1841:
1821:
1797:
1761:
1734:
1707:
1687:
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1628:
1598:
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1501:
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1347:
1237:
1209:
1171:
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1112:
1092:
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1038:
1010:
981:
859:
737:
666:
590:
555:
520:
493:
435:
405:
380:
353:
326:
303:
248:
205:
173:
151:
127:
100:
77:
5010: – Ring closed under countable intersections
3699:induced by the preorder. In other words, for all
3292:{\displaystyle (X,{\mathcal {T}},{\mathcal {F}})}
3049:. The complexes of a measurable space are called
2072:{\displaystyle S=f^{-1}=\{x\in X\mid f(x)\in B\}}
7707:
7692:{\displaystyle {\mathcal {F}}\neq \varnothing .}
5062: – Family of sets closed under intersection
3045:and the corresponding field of sets is called a
5113: – Relationship between certain categories
2629:{\displaystyle \mathbf {X} =(X,{\mathcal {F}})}
556:{\displaystyle \varnothing \in {\mathcal {F}}.}
5684:{\displaystyle \varnothing \in {\mathcal {F}}}
4167:{\displaystyle (X,(R_{i})_{I},{\mathcal {F}})}
3916:of the theory. They are a special case of the
5302:
3375:is a field of sets which is closed under the
3244:
2587:These definitions arise from considering the
2404:In other words: the atoms are a partition of
263:Fields of sets play an essential role in the
4929:{\displaystyle {\mathcal {C}}(\mathbf {X} )}
3885:
3842:
3802:
3759:
2507:and an example of a completion procedure in
2066:
2033:
5618:{\displaystyle A_{1}\cup A_{2}\cup \cdots }
5569:{\displaystyle A_{1}\cap A_{2}\cap \cdots }
3936:) if and only if it has a set of complexes
3202:and call its underlying measurable space a
2468:is the corresponding canonical surjection.
224:Fields of sets should not be confused with
5309:
5295:
5095: – Algebraic structure of set algebra
3206:. The points of a sample space are called
2526:Alternatively one can consider the set of
2483:can be constructed more generally for any
5651:{\displaystyle \Omega \in {\mathcal {F}}}
5396:
5171:
3532:Algebraic fields of sets and Stone fields
3612:{\displaystyle (X,\leq ,{\mathcal {F}})}
3021:Fields of sets with additional structure
2224:of the given field of sets, and defines
1100:itself as a field of sets. Elements of
7635:{\displaystyle A,B,A_{1},A_{2},\ldots }
4966:where a subset of a group was called a
3924:Algebraic and canonical preorder fields
3107:{\displaystyle (X,{\mathcal {F}},\mu )}
1578:
14:
7708:
3536:A topological field of sets is called
3491:(a formal mathematical abstraction of
7527:is a semiring where every complement
7255:{\displaystyle {\mathcal {F}}\colon }
5342:{\displaystyle {\mathcal {F}}\colon }
5089: – Function from sets to numbers
4230:i.e. a set with an indexed family of
2671:the corresponding topological space,
1636:(or, somewhat pedantically, the pair
4962:and has its origins in 19th century
4068:. The preorder fields obtained from
3431:forms a subalgebra of the power set
1417:Closed under countable intersections
591:{\displaystyle X\in {\mathcal {F}}.}
7220:
7213:
7175:
7154:
7147:
7140:
7121:
7114:
7083:
7055:
7048:
7041:
7025:
6930:
6851:
6832:
6794:
6787:
6767:
6760:
6753:
6723:
6716:
6688:
6681:
6674:
6655:
6648:
6641:
6634:
6627:
6620:
6613:
6606:
6599:
6577:
6570:
6549:
6542:
6535:
6528:
6521:
6499:
6485:
6478:
6464:
6457:
6450:
6443:
6424:
6403:
6389:
6382:
6375:
6368:
6349:
6314:
6307:
6300:
6293:
6225:
6218:
6211:
6189:
6182:
6144:
6130:
6101:
6080:
6013:
5981:
5927:
5878:
5871:
5847:
5798:
5791:
5719:
5712:
5159:Algebraic Polymodal Logic: A Survey
4009:{\displaystyle S\in {\mathcal {A}}}
3722:{\displaystyle S\in {\mathcal {F}}}
3467:{\displaystyle (X,{\mathcal {T}}).}
2354:{\displaystyle A\in {\mathcal {F}}}
2102:{\displaystyle S\in {\mathcal {F}}}
1069:{\displaystyle X\in {\mathcal {F}}}
414:that has the following properties:
236:. Similarly the term "algebra over
24:
7675:
7651:
7566:
7546:{\displaystyle \Omega \setminus A}
7534:
7500:
7389:
7345:
7276:
7244:
7227:
7206:
7168:
7161:
7128:
7076:
7069:
7062:
7009:
6994:
6987:
6980:
6973:
6970:
6967:
6960:
6953:
6946:
6914:
6899:
6892:
6885:
6878:
6875:
6872:
6865:
6858:
6816:
6801:
6780:
6777:
6774:
6737:
6730:
6709:
6702:
6695:
6662:
6584:
6563:
6556:
6506:
6492:
6471:
6431:
6417:
6410:
6396:
6356:
6342:
6335:
6328:
6321:
6274:
6267:
6260:
6253:
6246:
6239:
6232:
6196:
6137:
6094:
6087:
6059:
6052:
6045:
5974:
5967:
5960:
5953:
5946:
5934:
5920:
5913:
5906:
5899:
5892:
5885:
5854:
5840:
5833:
5826:
5819:
5812:
5805:
5775:
5768:
5761:
5754:
5747:
5740:
5733:
5726:
5676:
5643:
5635:
5520:{\displaystyle \Omega \setminus A}
5508:
5363:
5331:
5277:
5253:
4910:
4866:
4630:
4409:
4383:
4359:
4332:
4263:{\displaystyle (X,{\mathcal {F}})}
4252:
4156:
4112:. For this we consider structures
4001:
3945:
3826:
3823:
3743:
3740:
3737:
3714:
3684:{\displaystyle (X,{\mathcal {F}})}
3673:
3601:
3568:
3453:
3416:
3388:
3368:{\displaystyle (X,{\mathcal {F}})}
3357:
3328:{\displaystyle (X,{\mathcal {T}})}
3317:
3281:
3271:
3143:{\displaystyle (X,{\mathcal {F}})}
3132:
3090:
2960:
2874:
2716:
2700:{\displaystyle (X,{\mathcal {T}})}
2689:
2618:
2346:
2094:
1949:
1814:
1798:{\displaystyle (X,{\mathcal {F}})}
1787:
1550:
1494:
1442:
1396:
1340:
1288:
1230:
1210:{\displaystyle (X,{\mathcal {F}})}
1199:
1133:
1085:
1061:
1003:
971:
913:
849:
791:
727:
703:
656:
632:
580:
545:
483:
465:
346:
304:{\displaystyle (X,{\mathcal {F}})}
293:
120:
78:{\displaystyle (X,{\mathcal {F}})}
67:
25:
7732:
7683:
7537:
7471:
5511:
5482:
5208:
3986:if and only if for every complex
3026:Sigma algebras and measure spaces
870:Closed under finite intersections
454:
7221:
7214:
7207:
7176:
7169:
7162:
7155:
7148:
7141:
7122:
7115:
7084:
7077:
7070:
7063:
7056:
7049:
7042:
7026:
6988:
6981:
6974:
6961:
6954:
6947:
6931:
6893:
6886:
6879:
6866:
6859:
6852:
6833:
6795:
6788:
6781:
6768:
6761:
6754:
6738:
6731:
6724:
6717:
6710:
6703:
6696:
6689:
6682:
6675:
6656:
6649:
6642:
6635:
6628:
6621:
6614:
6607:
6600:
6578:
6571:
6564:
6557:
6550:
6543:
6536:
6529:
6522:
6500:
6493:
6486:
6479:
6472:
6465:
6458:
6451:
6444:
6425:
6418:
6411:
6404:
6397:
6390:
6383:
6376:
6369:
6350:
6343:
6336:
6329:
6322:
6315:
6308:
6301:
6294:
6275:
6268:
6261:
6254:
6247:
6240:
6233:
6226:
6219:
6212:
6190:
6183:
6138:
6131:
6095:
6088:
6081:
6060:
6053:
6046:
5975:
5968:
5961:
5954:
5947:
5928:
5921:
5914:
5907:
5900:
5893:
5886:
5879:
5872:
5848:
5841:
5834:
5827:
5820:
5813:
5806:
5799:
5792:
5776:
5769:
5762:
5755:
5748:
5741:
5734:
5727:
5720:
5713:
4919:
4368:
4286:
4278:) determined by a field of sets
2983:
2927:
2897:
2844:
2814:
2785:
2748:
2654:
2600:
2559:if and only if for every proper
1046:(with the same identity element
7577:{\displaystyle {\mathcal {F}}.}
7511:{\displaystyle {\mathcal {F}}.}
4219:{\displaystyle (X,(R_{i})_{I})}
4110:Boolean algebras with operators
3908:of a theory in the modal logic
2937:{\displaystyle T(\mathbf {X} )}
2854:{\displaystyle T(\mathbf {X} )}
2795:{\displaystyle T(\mathbf {X} )}
2758:{\displaystyle T(\mathbf {X} )}
2664:{\displaystyle T(\mathbf {X} )}
7659:{\displaystyle {\mathcal {F}}}
7284:{\displaystyle {\mathcal {F}}}
6168:{\displaystyle A_{i}\nearrow }
6162:
6037:{\displaystyle A_{i}\nearrow }
6031:
6005:{\displaystyle A_{i}\searrow }
5999:
5371:{\displaystyle {\mathcal {F}}}
5261:{\displaystyle {\mathcal {F}}}
5235:, Encyclopedia of Mathematics.
5195:Interior Algebras and Topology
5131:
5031:List of Boolean algebra topics
4923:
4915:
4874:{\displaystyle {\mathcal {F}}}
4829:
4791:
4693:
4661:
4444:
4435:
4421:
4378:
4372:
4364:
4337:
4293:
4257:
4241:
4213:
4204:
4190:
4181:
4161:
4142:
4128:
4119:
3953:{\displaystyle {\mathcal {A}}}
3836:
3830:
3753:
3747:
3678:
3662:
3638:
3626:
3606:
3584:
3562:McKinsey–Tarski Stone topology
3523:Given a topological space the
3458:
3442:
3424:{\displaystyle {\mathcal {F}}}
3396:{\displaystyle {\mathcal {T}}}
3362:
3346:
3322:
3306:
3286:
3260:
3137:
3121:
3101:
3079:
2968:{\displaystyle {\mathcal {F}}}
2931:
2923:
2882:{\displaystyle {\mathcal {F}}}
2848:
2840:
2789:
2781:
2752:
2744:
2724:{\displaystyle {\mathcal {T}}}
2694:
2678:
2658:
2650:
2623:
2607:
2260:
2254:
2057:
2051:
2027:
2021:
1957:{\displaystyle {\mathcal {F}}}
1924:
1875:
1856:
1822:{\displaystyle {\mathcal {F}}}
1792:
1776:
1662:
1643:
1238:{\displaystyle {\mathcal {F}}}
1204:
1188:
1141:{\displaystyle {\mathcal {F}}}
1093:{\displaystyle {\mathcal {F}}}
1011:{\displaystyle {\mathcal {F}}}
354:{\displaystyle {\mathcal {F}}}
298:
282:
270:
128:{\displaystyle {\mathcal {F}}}
72:
56:
13:
1:
5166:Varieties of complex algebras
5148:
2640:for a topology. We denote by
258:algebras over fields or rings
7480:{\displaystyle B\setminus A}
7421:{\displaystyle \varnothing }
6122:{\displaystyle A\subseteq B}
5491:{\displaystyle B\setminus A}
5403:{\displaystyle \,\supseteq }
4998: – mathematical concept
2990:{\displaystyle \mathbf {X} }
2904:{\displaystyle \mathbf {X} }
2821:{\displaystyle \mathbf {X} }
920: for all integers
798: for all integers
7:
7222:
7215:
7156:
7149:
7142:
7123:
7116:
7057:
7050:
7043:
7027:
6932:
6853:
6834:
6796:
6789:
6769:
6762:
6755:
6725:
6718:
6690:
6683:
6676:
6657:
6650:
6643:
6636:
6629:
6622:
6615:
6608:
6601:
6579:
6572:
6551:
6544:
6537:
6530:
6523:
6501:
6487:
6480:
6466:
6459:
6452:
6445:
6426:
6405:
6391:
6384:
6377:
6370:
6351:
6316:
6309:
6302:
6295:
6227:
6220:
6213:
6191:
6184:
6132:
6082:
5929:
5880:
5873:
5849:
5800:
5793:
5721:
5714:
5221:Encyclopedia of Mathematics
5021: – Algebraic structure
4973:
3928:A preorder field is called
10:
7737:
7642:are arbitrary elements of
7444:
7208:
7170:
7163:
7078:
7071:
7064:
6989:
6982:
6975:
6962:
6955:
6948:
6894:
6887:
6880:
6867:
6860:
6782:
6739:
6732:
6711:
6704:
6697:
6565:
6558:
6494:
6473:
6419:
6412:
6398:
6344:
6337:
6330:
6323:
6276:
6269:
6262:
6255:
6248:
6241:
6234:
6139:
6096:
6089:
6061:
6054:
6047:
5976:
5969:
5962:
5955:
5948:
5922:
5915:
5908:
5901:
5894:
5887:
5842:
5835:
5828:
5821:
5814:
5807:
5777:
5770:
5763:
5756:
5749:
5742:
5735:
5728:
5241:
4881:is the whole power set of
4093:Alexandrov bi-coreflection
3859: there exists a
3776: there exists a
3403:or equivalently under the
3245:Topological fields of sets
3150:is a measurable space and
2555:A field of sets is called
2544:A field of sets is called
2161:{\displaystyle B\subset Y}
2135:{\displaystyle B\in 2^{Y}}
748:Closed under finite unions
275:A field of sets is a pair
26:
5025:𝜆-system (Dynkin system)
4089:Alexandrov representation
3914:Lindenbaum–Tarski algebra
3644:{\displaystyle (X,\leq )}
3251:topological field of sets
3065:) and measurable spaces.
1881:{\displaystyle (Y,2^{Y})}
1668:{\displaystyle (Y,2^{Y})}
1076:). Many authors refer to
5124:
4569:is an operator of arity
4270:is a field of sets. The
2481:power set representation
2471:Similarly, every finite
2200:To this end one chooses
1933:{\displaystyle f:X\to Y}
7666:and it is assumed that
7461:where every complement
7395:{\displaystyle \Omega }
7351:{\displaystyle \Omega }
7235:Is necessarily true of
5462:{\displaystyle A\cup B}
5433:{\displaystyle A\cap B}
5322:Is necessarily true of
5283:{\displaystyle \Omega }
4954:(Historically the term
4718: there exist
4513:is a relation of arity
4479:{\displaystyle i\in I,}
3979:{\displaystyle x\leq y}
3063:abstract sigma algebras
2865:with compact open sets
1253:
1152:and are said to be the
7693:
7660:
7636:
7578:
7547:
7512:
7481:
7422:
7396:
7352:
7285:
7256:
7197:
7105:
7018:
6923:
6825:
6169:
6123:
6038:
6006:
5685:
5652:
5619:
5570:
5521:
5492:
5463:
5434:
5404:
5372:
5343:
5284:
5262:
4930:
4895:
4875:
4841:
4639:
4583:
4563:
4536:
4507:
4480:
4451:
4344:
4264:
4220:
4168:
4062:
4061:{\displaystyle y\in S}
4036:
4035:{\displaystyle x\in S}
4010:
3980:
3954:
3892:
3809:
3723:
3685:
3645:
3613:
3468:
3425:
3397:
3369:
3329:
3293:
3188:
3164:
3144:
3108:
2991:
2969:
2938:
2905:
2883:
2855:
2822:
2796:
2767:zero-dimensional space
2759:
2725:
2701:
2665:
2630:
2577:
2462:
2438:
2418:
2395:
2375:
2355:
2325:
2324:{\displaystyle x\in X}
2299:
2298:{\displaystyle x\in A}
2273:
2272:{\displaystyle f(x)=A}
2238:
2214:
2192:
2191:{\displaystyle 2^{n}.}
2162:
2136:
2103:
2073:
1985:
1958:
1934:
1908:; it means a function
1902:
1882:
1843:
1823:
1799:
1763:
1736:
1709:
1689:
1669:
1630:
1600:
1562:
1503:
1446:
1408:
1349:
1292:
1239:
1211:
1173:
1142:
1114:
1094:
1070:
1040:
1012:
983:
861:
739:
668:
592:
557:
522:
495:
437:
407:
382:
355:
328:
305:
250:
207:
175:
153:
129:
102:
79:
47:mathematical structure
7694:
7661:
7637:
7579:
7553:is equal to a finite
7548:
7513:
7487:is equal to a finite
7482:
7423:
7397:
7353:
7286:
7257:
7198:
7196:{\displaystyle \cap }
7106:
7104:{\displaystyle \cup }
7019:
6924:
6826:
6170:
6124:
6039:
6007:
5686:
5653:
5620:
5571:
5522:
5493:
5464:
5435:
5405:
5373:
5344:
5285:
5263:
4931:
4896:
4876:
4842:
4778: such that
4640:
4584:
4564:
4562:{\displaystyle f_{i}}
4537:
4508:
4506:{\displaystyle R_{i}}
4481:
4452:
4345:
4265:
4221:
4169:
4063:
4037:
4011:
3981:
3955:
3893:
3810:
3724:
3686:
3646:
3614:
3469:
3426:
3398:
3370:
3330:
3294:
3189:
3165:
3145:
3109:
2992:
2970:
2939:
2906:
2884:
2856:
2823:
2797:
2760:
2726:
2702:
2666:
2636:the complexes form a
2631:
2578:
2503:. It is the basis of
2463:
2444:is the corresponding
2439:
2419:
2401:cannot be a complex.
2396:
2376:
2356:
2326:
2300:
2274:
2239:
2220:to be the set of all
2215:
2193:
2163:
2137:
2104:
2074:
1986:
1984:{\displaystyle 2^{Y}}
1959:
1935:
1903:
1883:
1844:
1824:
1800:
1764:
1762:{\displaystyle 2^{n}}
1737:
1735:{\displaystyle 2^{Y}}
1710:
1690:
1670:
1631:
1629:{\displaystyle 2^{Y}}
1601:
1583:For an arbitrary set
1563:
1504:
1426:
1409:
1350:
1272:
1240:
1212:
1174:
1143:
1115:
1095:
1071:
1041:
1013:
984:
862:
740:
669:
593:
558:
523:
496:
438:
408:
383:
356:
329:
306:
265:representation theory
251:
208:
176:
154:
130:
103:
80:
49:consisting of a pair
7670:
7646:
7588:
7561:
7531:
7495:
7465:
7412:
7386:
7342:
7271:
7239:
7187:
7095:
6998:
6903:
6805:
6149:
6107:
6018:
5986:
5665:
5632:
5583:
5534:
5505:
5476:
5447:
5418:
5393:
5358:
5326:
5274:
5248:
4938:full complex algebra
4905:
4885:
4861:
4851:algebraic structures
4648:
4593:
4573:
4546:
4535:{\displaystyle n+1,}
4517:
4490:
4461:
4354:
4282:
4276:algebra of complexes
4238:
4228:relational structure
4178:
4116:
4046:
4020:
3990:
3964:
3940:
3918:general modal frames
3819:
3733:
3703:
3691:is a field of sets.
3659:
3623:
3581:
3558:Stone representation
3497:intuitionistic logic
3439:
3411:
3383:
3343:
3303:
3257:
3235:inner product spaces
3187:{\displaystyle \mu }
3178:
3163:{\displaystyle \mu }
3154:
3118:
3076:
2979:
2955:
2917:
2893:
2869:
2834:
2810:
2775:
2738:
2711:
2675:
2644:
2596:
2567:
2501:Stone representation
2452:
2428:
2408:
2385:
2365:
2335:
2309:
2283:
2248:
2228:
2204:
2172:
2146:
2113:
2083:
1999:
1968:
1944:
1912:
1892:
1853:
1833:
1809:
1773:
1746:
1719:
1699:
1679:
1640:
1613:
1587:
1579:Stone representation
1513:
1423:
1359:
1269:
1225:
1185:
1160:
1128:
1104:
1080:
1050:
1030:
998:
876:
754:
686:
677:Closed under binary
615:
606:Closed under binary
569:
534:
512:
448:
427:
394:
369:
341:
318:
279:
240:
194:
165:
143:
115:
92:
53:
5173:Johnstone, Peter T.
4981:Alexandrov topology
4234:defined on it, and
4085:Alexandrov topology
3697:Alexandrov topology
3518:intermediate logics
3482:algebraic semantics
3229:In applications to
3196:probability measure
1742:is finite (namely,
1695:is finite (namely,
934: and all
812: and all
710: for all
639: for all
472: for all
7689:
7656:
7632:
7574:
7543:
7508:
7477:
7418:
7392:
7348:
7281:
7252:
7193:
7101:
7014:
6919:
6821:
6165:
6119:
6034:
6002:
5681:
5648:
5615:
5566:
5517:
5488:
5459:
5430:
5400:
5368:
5339:
5280:
5258:
5075:Probability theory
4926:
4891:
4871:
4837:
4635:
4579:
4559:
4532:
4503:
4476:
4447:
4340:
4260:
4216:
4164:
4081:canonical preorder
4058:
4032:
4006:
3976:
3950:
3888:
3805:
3719:
3681:
3641:
3609:
3503:for these logics.
3464:
3421:
3393:
3365:
3325:
3289:
3239:topological groups
3224:probability theory
3184:
3174:defined on it. If
3160:
3140:
3104:
3041:), it is called a
3034:(hence also under
2987:
2965:
2934:
2901:
2879:
2851:
2818:
2792:
2755:
2721:
2697:
2661:
2626:
2573:
2458:
2434:
2414:
2391:
2371:
2351:
2321:
2295:
2269:
2234:
2210:
2188:
2158:
2132:
2099:
2069:
1981:
1954:
1930:
1898:
1878:
1839:
1819:
1795:
1759:
1732:
1705:
1685:
1665:
1626:
1599:{\displaystyle Y,}
1596:
1558:
1499:
1404:
1345:
1235:
1207:
1172:{\displaystyle X.}
1169:
1138:
1124:while elements of
1110:
1090:
1066:
1036:
1008:
979:
857:
735:
664:
588:
553:
518:
491:
433:
406:{\displaystyle X,}
403:
381:{\displaystyle X,}
378:
351:
324:
301:
246:
206:{\displaystyle X,}
203:
182:that contains the
171:
149:
125:
98:
75:
7704:
7703:
5216:"Algebra of sets"
4894:{\displaystyle X}
4779:
4719:
4582:{\displaystyle n}
3874:
3860:
3791:
3777:
3405:interior operator
3337:topological space
3200:probability space
2576:{\displaystyle X}
2490:Boolean algebra.
2461:{\displaystyle f}
2437:{\displaystyle Y}
2417:{\displaystyle X}
2394:{\displaystyle A}
2374:{\displaystyle A}
2237:{\displaystyle f}
2213:{\displaystyle Y}
1901:{\displaystyle Y}
1842:{\displaystyle X}
1708:{\displaystyle n}
1688:{\displaystyle Y}
1113:{\displaystyle X}
1039:{\displaystyle X}
1022:of the power set
935:
921:
813:
799:
711:
640:
521:{\displaystyle X}
473:
436:{\displaystyle X}
327:{\displaystyle X}
249:{\displaystyle X}
234:fields in physics
174:{\displaystyle X}
152:{\displaystyle X}
101:{\displaystyle X}
16:(Redirected from
7728:
7721:Families of sets
7698:
7696:
7695:
7690:
7679:
7678:
7665:
7663:
7662:
7657:
7655:
7654:
7641:
7639:
7638:
7633:
7625:
7624:
7612:
7611:
7583:
7581:
7580:
7575:
7570:
7569:
7552:
7550:
7549:
7544:
7517:
7515:
7514:
7509:
7504:
7503:
7486:
7484:
7483:
7478:
7458:
7446:Additionally, a
7440:
7429:
7428:
7427:
7425:
7424:
7419:
7403:
7402:
7401:
7399:
7398:
7393:
7377:
7376:
7368:
7367:
7359:
7358:
7357:
7355:
7354:
7349:
7331:
7330:
7322:
7321:
7313:
7312:
7304:
7295:
7294:
7290:
7288:
7287:
7282:
7280:
7279:
7263:
7262:
7261:
7259:
7258:
7253:
7248:
7247:
7225:
7224:
7218:
7217:
7211:
7210:
7204:
7202:
7200:
7199:
7194:
7183:(even arbitrary
7180:
7173:
7172:
7166:
7165:
7159:
7158:
7152:
7151:
7145:
7144:
7137:
7126:
7125:
7119:
7118:
7112:
7110:
7108:
7107:
7102:
7091:(even arbitrary
7088:
7081:
7080:
7074:
7073:
7067:
7066:
7060:
7059:
7053:
7052:
7046:
7045:
7038:
7030:
7029:
7023:
7021:
7020:
7015:
7013:
7012:
6992:
6991:
6985:
6984:
6978:
6977:
6965:
6964:
6958:
6957:
6951:
6950:
6943:
6935:
6934:
6928:
6926:
6925:
6920:
6918:
6917:
6897:
6896:
6890:
6889:
6883:
6882:
6870:
6869:
6863:
6862:
6856:
6855:
6848:
6845:
6837:
6836:
6830:
6828:
6827:
6822:
6820:
6819:
6799:
6798:
6792:
6791:
6785:
6784:
6772:
6771:
6765:
6764:
6758:
6757:
6750:
6742:
6741:
6735:
6734:
6728:
6727:
6721:
6720:
6714:
6713:
6707:
6706:
6700:
6699:
6693:
6692:
6686:
6685:
6679:
6678:
6671:
6660:
6659:
6653:
6652:
6646:
6645:
6639:
6638:
6632:
6631:
6625:
6624:
6618:
6617:
6611:
6610:
6604:
6603:
6596:
6593:
6582:
6581:
6575:
6574:
6568:
6567:
6561:
6560:
6554:
6553:
6547:
6546:
6540:
6539:
6533:
6532:
6526:
6525:
6518:
6517:
6504:
6503:
6497:
6496:
6490:
6489:
6483:
6482:
6476:
6475:
6469:
6468:
6462:
6461:
6455:
6454:
6448:
6447:
6440:
6429:
6428:
6422:
6421:
6415:
6414:
6408:
6407:
6401:
6400:
6394:
6393:
6387:
6386:
6380:
6379:
6373:
6372:
6365:
6354:
6353:
6347:
6346:
6340:
6339:
6333:
6332:
6326:
6325:
6319:
6318:
6312:
6311:
6305:
6304:
6298:
6297:
6290:
6289:
6288:(Measure theory)
6279:
6278:
6272:
6271:
6265:
6264:
6258:
6257:
6251:
6250:
6244:
6243:
6237:
6236:
6230:
6229:
6223:
6222:
6216:
6215:
6208:
6207:
6194:
6193:
6187:
6186:
6174:
6172:
6171:
6166:
6161:
6160:
6142:
6141:
6135:
6134:
6128:
6126:
6125:
6120:
6099:
6098:
6092:
6091:
6085:
6084:
6077:
6076:
6072:
6064:
6063:
6057:
6056:
6050:
6049:
6043:
6041:
6040:
6035:
6030:
6029:
6011:
6009:
6008:
6003:
5998:
5997:
5979:
5978:
5972:
5971:
5965:
5964:
5958:
5957:
5951:
5950:
5943:
5932:
5931:
5925:
5924:
5918:
5917:
5911:
5910:
5904:
5903:
5897:
5896:
5890:
5889:
5883:
5882:
5876:
5875:
5869:
5867:
5864:
5852:
5851:
5845:
5844:
5838:
5837:
5831:
5830:
5824:
5823:
5817:
5816:
5810:
5809:
5803:
5802:
5796:
5795:
5789:
5780:
5779:
5773:
5772:
5766:
5765:
5759:
5758:
5752:
5751:
5745:
5744:
5738:
5737:
5731:
5730:
5724:
5723:
5717:
5716:
5710:
5707:
5698:
5691:
5690:
5688:
5687:
5682:
5680:
5679:
5658:
5657:
5655:
5654:
5649:
5647:
5646:
5625:
5624:
5622:
5621:
5616:
5608:
5607:
5595:
5594:
5576:
5575:
5573:
5572:
5567:
5559:
5558:
5546:
5545:
5527:
5526:
5524:
5523:
5518:
5498:
5497:
5495:
5494:
5489:
5469:
5468:
5466:
5465:
5460:
5440:
5439:
5437:
5436:
5431:
5411:
5409:
5407:
5406:
5401:
5382:
5381:
5377:
5375:
5374:
5369:
5367:
5366:
5350:
5349:
5348:
5346:
5345:
5340:
5335:
5334:
5311:
5304:
5297:
5290:
5289:
5287:
5286:
5281:
5267:
5265:
5264:
5259:
5257:
5256:
5239:
5238:
5229:
5204:, Elsevier, 2006
5193:Naturman, C.A.,
5190:
5142:
5139:De Morgan's laws
5135:
5071:
5066:Preordered field
5059:
5053:
5050:
5041:
5019:Interior algebra
5007:
5001:
4986:
4935:
4933:
4932:
4927:
4922:
4914:
4913:
4900:
4898:
4897:
4892:
4880:
4878:
4877:
4872:
4870:
4869:
4846:
4844:
4843:
4838:
4836:
4832:
4822:
4821:
4803:
4802:
4790:
4789:
4780:
4777:
4775:
4774:
4762:
4761:
4743:
4742:
4730:
4729:
4720:
4717:
4692:
4691:
4673:
4672:
4660:
4659:
4644:
4642:
4641:
4636:
4634:
4633:
4624:
4623:
4605:
4604:
4588:
4586:
4585:
4580:
4568:
4566:
4565:
4560:
4558:
4557:
4541:
4539:
4538:
4533:
4512:
4510:
4509:
4504:
4502:
4501:
4485:
4483:
4482:
4477:
4456:
4454:
4453:
4448:
4443:
4442:
4433:
4432:
4387:
4386:
4371:
4363:
4362:
4349:
4347:
4346:
4341:
4336:
4335:
4326:
4325:
4320:
4316:
4315:
4289:
4269:
4267:
4266:
4261:
4256:
4255:
4225:
4223:
4222:
4217:
4212:
4211:
4202:
4201:
4173:
4171:
4170:
4165:
4160:
4159:
4150:
4149:
4140:
4139:
4067:
4065:
4064:
4059:
4041:
4039:
4038:
4033:
4015:
4013:
4012:
4007:
4005:
4004:
3985:
3983:
3982:
3977:
3959:
3957:
3956:
3951:
3949:
3948:
3906:Kripke semantics
3897:
3895:
3894:
3889:
3875:
3873: with
3872:
3861:
3858:
3829:
3814:
3812:
3811:
3806:
3792:
3790: with
3789:
3778:
3775:
3746:
3728:
3726:
3725:
3720:
3718:
3717:
3690:
3688:
3687:
3682:
3677:
3676:
3650:
3648:
3647:
3642:
3618:
3616:
3615:
3610:
3605:
3604:
3514:modal companions
3478:Heyting algebras
3473:
3471:
3470:
3465:
3457:
3456:
3433:interior algebra
3430:
3428:
3427:
3422:
3420:
3419:
3402:
3400:
3399:
3394:
3392:
3391:
3377:closure operator
3374:
3372:
3371:
3366:
3361:
3360:
3334:
3332:
3331:
3326:
3321:
3320:
3298:
3296:
3295:
3290:
3285:
3284:
3275:
3274:
3193:
3191:
3190:
3185:
3169:
3167:
3166:
3161:
3149:
3147:
3146:
3141:
3136:
3135:
3113:
3111:
3110:
3105:
3094:
3093:
3047:measurable space
2996:
2994:
2993:
2988:
2986:
2974:
2972:
2971:
2966:
2964:
2963:
2943:
2941:
2940:
2935:
2930:
2910:
2908:
2907:
2902:
2900:
2888:
2886:
2885:
2880:
2878:
2877:
2860:
2858:
2857:
2852:
2847:
2827:
2825:
2824:
2819:
2817:
2801:
2799:
2798:
2793:
2788:
2764:
2762:
2761:
2756:
2751:
2730:
2728:
2727:
2722:
2720:
2719:
2706:
2704:
2703:
2698:
2693:
2692:
2670:
2668:
2667:
2662:
2657:
2635:
2633:
2632:
2627:
2622:
2621:
2603:
2582:
2580:
2579:
2574:
2467:
2465:
2464:
2459:
2443:
2441:
2440:
2435:
2423:
2421:
2420:
2415:
2400:
2398:
2397:
2392:
2380:
2378:
2377:
2372:
2360:
2358:
2357:
2352:
2350:
2349:
2330:
2328:
2327:
2322:
2304:
2302:
2301:
2296:
2278:
2276:
2275:
2270:
2243:
2241:
2240:
2235:
2219:
2217:
2216:
2211:
2197:
2195:
2194:
2189:
2184:
2183:
2167:
2165:
2164:
2159:
2141:
2139:
2138:
2133:
2131:
2130:
2108:
2106:
2105:
2100:
2098:
2097:
2078:
2076:
2075:
2070:
2020:
2019:
1990:
1988:
1987:
1982:
1980:
1979:
1963:
1961:
1960:
1955:
1953:
1952:
1939:
1937:
1936:
1931:
1907:
1905:
1904:
1899:
1887:
1885:
1884:
1879:
1874:
1873:
1848:
1846:
1845:
1840:
1828:
1826:
1825:
1820:
1818:
1817:
1804:
1802:
1801:
1796:
1791:
1790:
1768:
1766:
1765:
1760:
1758:
1757:
1741:
1739:
1738:
1733:
1731:
1730:
1715:-element), then
1714:
1712:
1711:
1706:
1694:
1692:
1691:
1686:
1674:
1672:
1671:
1666:
1661:
1660:
1635:
1633:
1632:
1627:
1625:
1624:
1605:
1603:
1602:
1597:
1567:
1565:
1564:
1559:
1554:
1553:
1538:
1537:
1525:
1524:
1508:
1506:
1505:
1500:
1498:
1497:
1482:
1481:
1469:
1468:
1456:
1455:
1445:
1440:
1413:
1411:
1410:
1405:
1400:
1399:
1384:
1383:
1371:
1370:
1354:
1352:
1351:
1346:
1344:
1343:
1328:
1327:
1315:
1314:
1302:
1301:
1291:
1286:
1244:
1242:
1241:
1236:
1234:
1233:
1221:and the algebra
1216:
1214:
1213:
1208:
1203:
1202:
1181:A field of sets
1178:
1176:
1175:
1170:
1147:
1145:
1144:
1139:
1137:
1136:
1119:
1117:
1116:
1111:
1099:
1097:
1096:
1091:
1089:
1088:
1075:
1073:
1072:
1067:
1065:
1064:
1045:
1043:
1042:
1037:
1017:
1015:
1014:
1009:
1007:
1006:
994:In other words,
988:
986:
985:
980:
975:
974:
965:
964:
946:
945:
936:
933:
922:
919:
917:
916:
907:
906:
888:
887:
866:
864:
863:
858:
853:
852:
843:
842:
824:
823:
814:
811:
800:
797:
795:
794:
785:
784:
766:
765:
744:
742:
741:
736:
731:
730:
712:
709:
707:
706:
673:
671:
670:
665:
660:
659:
641:
638:
636:
635:
597:
595:
594:
589:
584:
583:
562:
560:
559:
554:
549:
548:
527:
525:
524:
519:
500:
498:
497:
492:
487:
486:
474:
471:
469:
468:
442:
440:
439:
434:
412:
410:
409:
404:
387:
385:
384:
379:
360:
358:
357:
352:
350:
349:
333:
331:
330:
325:
311:consisting of a
310:
308:
307:
302:
297:
296:
260:in ring theory.
255:
253:
252:
247:
212:
210:
209:
204:
180:
178:
177:
172:
158:
156:
155:
150:
134:
132:
131:
126:
124:
123:
107:
105:
104:
99:
85:consisting of a
84:
82:
81:
76:
71:
70:
21:
7736:
7735:
7731:
7730:
7729:
7727:
7726:
7725:
7716:Boolean algebra
7706:
7705:
7699:
7674:
7673:
7671:
7668:
7667:
7650:
7649:
7647:
7644:
7643:
7620:
7616:
7607:
7603:
7589:
7586:
7585:
7584:
7565:
7564:
7562:
7559:
7558:
7532:
7529:
7528:
7518:
7499:
7498:
7496:
7493:
7492:
7466:
7463:
7462:
7456:
7437:
7435:
7432:
7413:
7410:
7409:
7407:
7406:
7387:
7384:
7383:
7381:
7380:
7374:
7372:
7371:
7365:
7363:
7362:
7343:
7340:
7339:
7337:
7335:
7334:
7328:
7326:
7325:
7319:
7317:
7316:
7310:
7308:
7307:
7301:
7298:
7275:
7274:
7272:
7269:
7268:
7266:
7265:
7264:
7243:
7242:
7240:
7237:
7236:
7234:
7233:
7188:
7185:
7184:
7182:
7181:
7136:Closed Topology
7135:
7096:
7093:
7092:
7090:
7089:
7036:
7008:
7007:
6999:
6996:
6995:
6941:
6913:
6912:
6904:
6901:
6900:
6846:
6843:
6815:
6814:
6806:
6803:
6802:
6748:
6669:
6594:
6591:
6515:
6513:
6438:
6363:
6287:
6285:
6205:
6203:
6176:
6156:
6152:
6150:
6147:
6146:
6108:
6105:
6104:
6103:
6075:(Dynkin System)
6074:
6073:
6070:
6025:
6021:
6019:
6016:
6015:
5993:
5989:
5987:
5984:
5983:
5941:
5865:
5862:
5860:
5785:
5705:
5703:
5694:
5675:
5674:
5666:
5663:
5662:
5661:
5642:
5641:
5633:
5630:
5629:
5628:
5603:
5599:
5590:
5586:
5584:
5581:
5580:
5579:
5554:
5550:
5541:
5537:
5535:
5532:
5531:
5530:
5506:
5503:
5502:
5501:
5477:
5474:
5473:
5472:
5448:
5445:
5444:
5443:
5419:
5416:
5415:
5414:
5394:
5391:
5390:
5388:
5385:
5362:
5361:
5359:
5356:
5355:
5353:
5352:
5351:
5330:
5329:
5327:
5324:
5323:
5321:
5320:
5315:
5275:
5272:
5271:
5252:
5251:
5249:
5246:
5245:
5242:
5233:Algebra of sets
5214:
5211:
5187:
5164:Goldblatt, R.,
5151:
5146:
5145:
5136:
5132:
5127:
5122:
5069:
5057:
5051:
5048:
5047: – theorem
5039:
5005:
4999:
4990:Algebra of sets
4984:
4976:
4918:
4909:
4908:
4906:
4903:
4902:
4886:
4883:
4882:
4865:
4864:
4862:
4859:
4858:
4817:
4813:
4798:
4794:
4785:
4781:
4776:
4770:
4766:
4757:
4753:
4738:
4734:
4725:
4721:
4716:
4703:
4699:
4687:
4683:
4668:
4664:
4655:
4651:
4649:
4646:
4645:
4629:
4628:
4619:
4615:
4600:
4596:
4594:
4591:
4590:
4574:
4571:
4570:
4553:
4549:
4547:
4544:
4543:
4518:
4515:
4514:
4497:
4493:
4491:
4488:
4487:
4462:
4459:
4458:
4438:
4434:
4428:
4424:
4382:
4381:
4367:
4358:
4357:
4355:
4352:
4351:
4331:
4330:
4321:
4311:
4307:
4303:
4302:
4285:
4283:
4280:
4279:
4272:complex algebra
4251:
4250:
4239:
4236:
4235:
4207:
4203:
4197:
4193:
4179:
4176:
4175:
4155:
4154:
4145:
4141:
4135:
4131:
4117:
4114:
4113:
4106:
4047:
4044:
4043:
4021:
4018:
4017:
4000:
3999:
3991:
3988:
3987:
3965:
3962:
3961:
3944:
3943:
3941:
3938:
3937:
3926:
3902:possible worlds
3871:
3857:
3822:
3820:
3817:
3816:
3788:
3774:
3736:
3734:
3731:
3730:
3713:
3712:
3704:
3701:
3700:
3672:
3671:
3660:
3657:
3656:
3624:
3621:
3620:
3600:
3599:
3582:
3579:
3578:
3571:
3569:Preorder fields
3534:
3509:Heyting algebra
3493:epistemic logic
3452:
3451:
3440:
3437:
3436:
3415:
3414:
3412:
3409:
3408:
3387:
3386:
3384:
3381:
3380:
3356:
3355:
3344:
3341:
3340:
3316:
3315:
3304:
3301:
3300:
3280:
3279:
3270:
3269:
3258:
3255:
3254:
3247:
3179:
3176:
3175:
3155:
3152:
3151:
3131:
3130:
3119:
3116:
3115:
3089:
3088:
3077:
3074:
3073:
3051:measurable sets
3028:
3023:
2982:
2980:
2977:
2976:
2975:if and only if
2959:
2958:
2956:
2953:
2952:
2926:
2918:
2915:
2914:
2896:
2894:
2891:
2890:
2889:if and only if
2873:
2872:
2870:
2867:
2866:
2843:
2835:
2832:
2831:
2813:
2811:
2808:
2807:
2806:if and only if
2804:Hausdorff space
2784:
2776:
2773:
2772:
2747:
2739:
2736:
2735:
2715:
2714:
2712:
2709:
2708:
2688:
2687:
2676:
2673:
2672:
2653:
2645:
2642:
2641:
2617:
2616:
2599:
2597:
2594:
2593:
2568:
2565:
2564:
2541:
2473:Boolean algebra
2453:
2450:
2449:
2429:
2426:
2425:
2409:
2406:
2405:
2386:
2383:
2382:
2381:different from
2366:
2363:
2362:
2345:
2344:
2336:
2333:
2332:
2310:
2307:
2306:
2284:
2281:
2280:
2249:
2246:
2245:
2229:
2226:
2225:
2205:
2202:
2201:
2179:
2175:
2173:
2170:
2169:
2147:
2144:
2143:
2126:
2122:
2114:
2111:
2110:
2093:
2092:
2084:
2081:
2080:
2012:
2008:
2000:
1997:
1996:
1975:
1971:
1969:
1966:
1965:
1948:
1947:
1945:
1942:
1941:
1913:
1910:
1909:
1893:
1890:
1889:
1869:
1865:
1854:
1851:
1850:
1834:
1831:
1830:
1813:
1812:
1810:
1807:
1806:
1786:
1785:
1774:
1771:
1770:
1753:
1749:
1747:
1744:
1743:
1726:
1722:
1720:
1717:
1716:
1700:
1697:
1696:
1680:
1677:
1676:
1656:
1652:
1641:
1638:
1637:
1620:
1616:
1614:
1611:
1610:
1588:
1585:
1584:
1581:
1576:
1549:
1548:
1533:
1529:
1520:
1516:
1514:
1511:
1510:
1493:
1492:
1477:
1473:
1464:
1460:
1451:
1447:
1441:
1430:
1424:
1421:
1420:
1395:
1394:
1379:
1375:
1366:
1362:
1360:
1357:
1356:
1339:
1338:
1323:
1319:
1310:
1306:
1297:
1293:
1287:
1276:
1270:
1267:
1266:
1229:
1228:
1226:
1223:
1222:
1219:σ-field of sets
1198:
1197:
1186:
1183:
1182:
1161:
1158:
1157:
1154:admissible sets
1132:
1131:
1129:
1126:
1125:
1105:
1102:
1101:
1084:
1083:
1081:
1078:
1077:
1060:
1059:
1051:
1048:
1047:
1031:
1028:
1027:
1024:Boolean algebra
1002:
1001:
999:
996:
995:
970:
969:
960:
956:
941:
937:
932:
918:
912:
911:
902:
898:
883:
879:
877:
874:
873:
848:
847:
838:
834:
819:
815:
810:
796:
790:
789:
780:
776:
761:
757:
755:
752:
751:
726:
725:
708:
702:
701:
687:
684:
683:
655:
654:
637:
631:
630:
616:
613:
612:
579:
578:
570:
567:
566:
544:
543:
535:
532:
531:
530:as an element:
513:
510:
509:
482:
481:
470:
464:
463:
449:
446:
445:
428:
425:
424:
421:complementation
395:
392:
391:
370:
367:
366:
345:
344:
342:
339:
338:
319:
316:
315:
292:
291:
280:
277:
276:
273:
241:
238:
237:
195:
192:
191:
166:
163:
162:
144:
141:
140:
119:
118:
116:
113:
112:
93:
90:
89:
66:
65:
54:
51:
50:
35:
32:
29:Algebra of sets
23:
22:
15:
12:
11:
5:
7734:
7724:
7723:
7718:
7702:
7701:
7688:
7685:
7682:
7677:
7653:
7631:
7628:
7623:
7619:
7615:
7610:
7606:
7602:
7599:
7596:
7593:
7573:
7568:
7555:disjoint union
7542:
7539:
7536:
7525:
7507:
7502:
7489:disjoint union
7476:
7473:
7470:
7452:
7442:
7441:
7430:
7417:
7404:
7391:
7378:
7369:
7360:
7347:
7332:
7323:
7314:
7305:
7296:
7278:
7251:
7246:
7230:
7229:
7226:
7219:
7212:
7205:
7192:
7174:
7167:
7160:
7153:
7146:
7139:
7131:
7130:
7127:
7120:
7113:
7100:
7082:
7075:
7068:
7061:
7054:
7047:
7040:
7032:
7031:
7024:
7011:
7006:
7003:
6993:
6986:
6979:
6972:
6969:
6966:
6959:
6952:
6945:
6942:Filter subbase
6937:
6936:
6929:
6916:
6911:
6908:
6898:
6891:
6884:
6877:
6874:
6871:
6864:
6857:
6850:
6839:
6838:
6831:
6818:
6813:
6810:
6800:
6793:
6786:
6779:
6776:
6773:
6766:
6759:
6752:
6744:
6743:
6736:
6729:
6722:
6715:
6708:
6701:
6694:
6687:
6680:
6673:
6665:
6664:
6661:
6654:
6647:
6640:
6633:
6626:
6619:
6612:
6605:
6598:
6587:
6586:
6583:
6576:
6569:
6562:
6555:
6548:
6541:
6534:
6527:
6520:
6509:
6508:
6505:
6498:
6491:
6484:
6477:
6470:
6463:
6456:
6449:
6442:
6434:
6433:
6430:
6423:
6416:
6409:
6402:
6395:
6388:
6381:
6374:
6367:
6359:
6358:
6355:
6348:
6341:
6334:
6327:
6320:
6313:
6306:
6299:
6292:
6281:
6280:
6273:
6266:
6259:
6252:
6245:
6238:
6231:
6224:
6217:
6210:
6206:(Order theory)
6199:
6198:
6195:
6188:
6181:
6164:
6159:
6155:
6143:
6136:
6129:
6118:
6115:
6112:
6100:
6093:
6086:
6079:
6066:
6065:
6058:
6051:
6044:
6033:
6028:
6024:
6012:
6001:
5996:
5992:
5980:
5973:
5966:
5959:
5952:
5945:
5942:Monotone class
5937:
5936:
5933:
5926:
5919:
5912:
5905:
5898:
5891:
5884:
5877:
5870:
5857:
5856:
5853:
5846:
5839:
5832:
5825:
5818:
5811:
5804:
5797:
5790:
5782:
5781:
5774:
5767:
5760:
5753:
5746:
5739:
5732:
5725:
5718:
5711:
5700:
5699:
5692:
5678:
5673:
5670:
5659:
5645:
5640:
5637:
5626:
5614:
5611:
5606:
5602:
5598:
5593:
5589:
5577:
5565:
5562:
5557:
5553:
5549:
5544:
5540:
5528:
5516:
5513:
5510:
5499:
5487:
5484:
5481:
5470:
5458:
5455:
5452:
5441:
5429:
5426:
5423:
5412:
5399:
5383:
5365:
5338:
5333:
5317:
5316:
5314:
5313:
5306:
5299:
5291:
5279:
5255:
5237:
5236:
5230:
5210:
5209:External links
5207:
5206:
5205:
5198:
5191:
5185:
5169:
5162:
5150:
5147:
5144:
5143:
5129:
5128:
5126:
5123:
5121:
5120:
5114:
5108:
5102:
5096:
5090:
5084:
5078:
5072:
5063:
5054:
5045:Monotone class
5042:
5036:Measure theory
5033:
5028:
5022:
5016:
5011:
5002:
4993:
4987:
4977:
4975:
4972:
4925:
4921:
4917:
4912:
4890:
4868:
4835:
4831:
4828:
4825:
4820:
4816:
4812:
4809:
4806:
4801:
4797:
4793:
4788:
4784:
4773:
4769:
4765:
4760:
4756:
4752:
4749:
4746:
4741:
4737:
4733:
4728:
4724:
4715:
4712:
4709:
4706:
4702:
4698:
4695:
4690:
4686:
4682:
4679:
4676:
4671:
4667:
4663:
4658:
4654:
4632:
4627:
4622:
4618:
4614:
4611:
4608:
4603:
4599:
4578:
4556:
4552:
4531:
4528:
4525:
4522:
4500:
4496:
4475:
4472:
4469:
4466:
4457:where for all
4446:
4441:
4437:
4431:
4427:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4385:
4380:
4377:
4374:
4370:
4366:
4361:
4339:
4334:
4329:
4324:
4319:
4314:
4310:
4306:
4301:
4298:
4295:
4292:
4288:
4259:
4254:
4249:
4246:
4243:
4215:
4210:
4206:
4200:
4196:
4192:
4189:
4186:
4183:
4163:
4158:
4153:
4148:
4144:
4138:
4134:
4130:
4127:
4124:
4121:
4105:
4102:
4091:" is just the
4057:
4054:
4051:
4031:
4028:
4025:
4003:
3998:
3995:
3975:
3972:
3969:
3947:
3925:
3922:
3887:
3884:
3881:
3878:
3870:
3867:
3864:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3828:
3825:
3804:
3801:
3798:
3795:
3787:
3784:
3781:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3745:
3742:
3739:
3716:
3711:
3708:
3680:
3675:
3670:
3667:
3664:
3653:preordered set
3640:
3637:
3634:
3631:
3628:
3608:
3603:
3598:
3595:
3592:
3589:
3586:
3575:preorder field
3570:
3567:
3533:
3530:
3463:
3460:
3455:
3450:
3447:
3444:
3418:
3390:
3364:
3359:
3354:
3351:
3348:
3324:
3319:
3314:
3311:
3308:
3288:
3283:
3278:
3273:
3268:
3265:
3262:
3246:
3243:
3226:respectively.
3220:measure theory
3198:we speak of a
3183:
3159:
3139:
3134:
3129:
3126:
3123:
3103:
3100:
3097:
3092:
3087:
3084:
3081:
3027:
3024:
3022:
3019:
3003:
3002:
2985:
2962:
2933:
2929:
2925:
2922:
2912:
2899:
2876:
2850:
2846:
2842:
2839:
2829:
2828:is separative.
2816:
2791:
2787:
2783:
2780:
2770:
2754:
2750:
2746:
2743:
2718:
2696:
2691:
2686:
2683:
2680:
2660:
2656:
2652:
2649:
2625:
2620:
2615:
2612:
2609:
2606:
2602:
2585:
2584:
2572:
2553:
2550:differentiated
2540:
2537:
2457:
2433:
2413:
2390:
2370:
2348:
2343:
2340:
2331:and a complex
2320:
2317:
2314:
2294:
2291:
2288:
2268:
2265:
2262:
2259:
2256:
2253:
2233:
2209:
2187:
2182:
2178:
2157:
2154:
2151:
2129:
2125:
2121:
2118:
2096:
2091:
2088:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2018:
2015:
2011:
2007:
2004:
1978:
1974:
1951:
1929:
1926:
1923:
1920:
1917:
1897:
1877:
1872:
1868:
1864:
1861:
1858:
1838:
1829:finite, while
1816:
1794:
1789:
1784:
1781:
1778:
1756:
1752:
1729:
1725:
1704:
1684:
1664:
1659:
1655:
1651:
1648:
1645:
1623:
1619:
1595:
1592:
1580:
1577:
1575:
1572:
1571:
1570:
1569:
1568:
1557:
1552:
1547:
1544:
1541:
1536:
1532:
1528:
1523:
1519:
1496:
1491:
1488:
1485:
1480:
1476:
1472:
1467:
1463:
1459:
1454:
1450:
1444:
1439:
1436:
1433:
1429:
1418:
1414:
1403:
1398:
1393:
1390:
1387:
1382:
1378:
1374:
1369:
1365:
1342:
1337:
1334:
1331:
1326:
1322:
1318:
1313:
1309:
1305:
1300:
1296:
1290:
1285:
1282:
1279:
1275:
1264:
1232:
1206:
1201:
1196:
1193:
1190:
1168:
1165:
1135:
1109:
1087:
1063:
1058:
1055:
1035:
1005:
992:
991:
990:
989:
978:
973:
968:
963:
959:
955:
952:
949:
944:
940:
931:
928:
925:
915:
910:
905:
901:
897:
894:
891:
886:
882:
871:
867:
856:
851:
846:
841:
837:
833:
830:
827:
822:
818:
809:
806:
803:
793:
788:
783:
779:
775:
772:
769:
764:
760:
749:
745:
734:
729:
724:
721:
718:
715:
705:
700:
697:
694:
691:
681:
674:
663:
658:
653:
650:
647:
644:
634:
629:
626:
623:
620:
610:
600:
599:
598:
587:
582:
577:
574:
552:
547:
542:
539:
529:
517:
501:
490:
485:
480:
477:
467:
462:
459:
456:
453:
443:
432:
402:
399:
377:
374:
348:
323:
300:
295:
290:
287:
284:
272:
269:
245:
202:
199:
170:
148:
122:
97:
74:
69:
64:
61:
58:
33:
9:
6:
4:
3:
2:
7733:
7722:
7719:
7717:
7714:
7713:
7711:
7700:
7686:
7680:
7629:
7626:
7621:
7617:
7613:
7608:
7604:
7600:
7597:
7594:
7591:
7571:
7556:
7540:
7526:
7524:
7521:
7505:
7490:
7474:
7468:
7460:
7453:
7451:
7448:
7443:
7439:
7431:
7415:
7405:
7379:
7370:
7366:intersections
7361:
7333:
7324:
7315:
7311:intersections
7306:
7303:
7297:
7293:
7292:closed under:
7249:
7232:
7231:
7190:
7179:
7138:
7133:
7132:
7098:
7087:
7039:
7037:Open Topology
7034:
7033:
7004:
7001:
6944:
6939:
6938:
6909:
6906:
6849:
6847:(Filter base)
6841:
6840:
6811:
6808:
6751:
6746:
6745:
6672:
6667:
6666:
6597:
6589:
6588:
6519:
6511:
6510:
6441:
6436:
6435:
6366:
6361:
6360:
6291:
6283:
6282:
6209:
6201:
6200:
6180:
6157:
6153:
6116:
6113:
6110:
6078:
6068:
6067:
6026:
6022:
5994:
5990:
5944:
5939:
5938:
5868:
5859:
5858:
5788:
5784:
5783:
5709:
5702:
5701:
5697:
5693:
5671:
5668:
5660:
5638:
5627:
5612:
5609:
5604:
5600:
5596:
5591:
5587:
5578:
5563:
5560:
5555:
5551:
5547:
5542:
5538:
5529:
5514:
5500:
5485:
5479:
5471:
5456:
5453:
5450:
5442:
5427:
5424:
5421:
5413:
5410:
5397:
5384:
5380:
5379:closed under:
5336:
5319:
5318:
5312:
5307:
5305:
5300:
5298:
5293:
5292:
5269:
5240:
5234:
5231:
5227:
5223:
5222:
5217:
5213:
5212:
5203:
5199:
5196:
5192:
5188:
5186:0-521-33779-8
5182:
5178:
5174:
5170:
5167:
5163:
5160:
5156:
5155:Goldblatt, R.
5153:
5152:
5140:
5134:
5130:
5118:
5115:
5112:
5111:Stone duality
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5085:
5082:
5079:
5076:
5073:
5067:
5064:
5061:
5055:
5046:
5043:
5037:
5034:
5032:
5029:
5026:
5023:
5020:
5017:
5015:
5014:General frame
5012:
5009:
5003:
4997:
4994:
4991:
4988:
4982:
4979:
4978:
4971:
4969:
4965:
4961:
4957:
4952:
4950:
4945:
4943:
4942:power algebra
4939:
4888:
4856:
4852:
4847:
4833:
4826:
4823:
4818:
4814:
4810:
4807:
4804:
4799:
4795:
4786:
4782:
4771:
4767:
4763:
4758:
4754:
4750:
4747:
4744:
4739:
4735:
4731:
4726:
4722:
4713:
4710:
4707:
4704:
4700:
4696:
4688:
4684:
4680:
4677:
4674:
4669:
4665:
4656:
4652:
4625:
4620:
4616:
4612:
4609:
4606:
4601:
4597:
4576:
4554:
4550:
4529:
4526:
4523:
4520:
4498:
4494:
4473:
4470:
4467:
4464:
4439:
4429:
4425:
4418:
4415:
4412:
4406:
4400:
4397:
4394:
4391:
4388:
4375:
4327:
4322:
4317:
4312:
4308:
4304:
4299:
4296:
4290:
4277:
4273:
4247:
4244:
4233:
4229:
4208:
4198:
4194:
4187:
4184:
4151:
4146:
4136:
4132:
4125:
4122:
4111:
4101:
4099:
4094:
4090:
4086:
4082:
4078:
4073:
4071:
4055:
4052:
4049:
4029:
4026:
4023:
3996:
3993:
3973:
3970:
3967:
3935:
3931:
3921:
3919:
3915:
3911:
3907:
3903:
3898:
3882:
3879:
3876:
3868:
3865:
3862:
3854:
3851:
3848:
3845:
3839:
3833:
3799:
3796:
3793:
3785:
3782:
3779:
3771:
3768:
3765:
3762:
3756:
3750:
3709:
3706:
3698:
3692:
3668:
3665:
3654:
3635:
3632:
3629:
3596:
3593:
3590:
3587:
3576:
3566:
3563:
3559:
3555:
3554:open elements
3550:
3545:
3541:
3539:
3529:
3526:
3521:
3519:
3515:
3510:
3504:
3502:
3498:
3494:
3490:
3487:
3483:
3479:
3474:
3461:
3448:
3445:
3434:
3406:
3378:
3352:
3349:
3338:
3312:
3309:
3276:
3266:
3263:
3252:
3242:
3240:
3236:
3232:
3227:
3225:
3221:
3217:
3213:
3209:
3208:sample points
3205:
3201:
3197:
3194:is in fact a
3181:
3173:
3157:
3127:
3124:
3098:
3095:
3085:
3082:
3071:
3070:measure space
3066:
3064:
3060:
3056:
3052:
3048:
3044:
3043:sigma algebra
3040:
3039:intersections
3037:
3033:
3018:
3016:
3012:
3011:Stone duality
3008:
3000:
2951:
2947:
2946:Boolean space
2920:
2913:
2864:
2863:compact space
2837:
2830:
2805:
2778:
2771:
2768:
2741:
2734:
2733:
2732:
2684:
2681:
2647:
2639:
2613:
2610:
2604:
2590:
2570:
2562:
2558:
2554:
2551:
2547:
2543:
2542:
2536:
2534:
2529:
2528:homomorphisms
2524:
2522:
2521:Dedekind cuts
2519:, similar to
2518:
2514:
2510:
2506:
2502:
2497:
2491:
2489:
2486:
2482:
2478:
2474:
2469:
2455:
2447:
2431:
2411:
2402:
2388:
2368:
2341:
2338:
2318:
2315:
2312:
2292:
2289:
2286:
2266:
2263:
2257:
2251:
2231:
2223:
2207:
2198:
2185:
2180:
2176:
2155:
2152:
2149:
2127:
2123:
2119:
2116:
2089:
2086:
2063:
2060:
2054:
2048:
2045:
2042:
2039:
2036:
2030:
2024:
2016:
2013:
2009:
2005:
2002:
1994:
1993:inverse image
1976:
1972:
1927:
1921:
1918:
1915:
1895:
1870:
1866:
1862:
1859:
1836:
1782:
1779:
1754:
1750:
1727:
1723:
1702:
1682:
1657:
1653:
1649:
1646:
1621:
1617:
1609:
1593:
1590:
1555:
1545:
1542:
1539:
1534:
1530:
1526:
1521:
1517:
1489:
1486:
1483:
1478:
1474:
1470:
1465:
1461:
1457:
1452:
1448:
1437:
1434:
1431:
1427:
1416:
1415:
1401:
1391:
1388:
1385:
1380:
1376:
1372:
1367:
1363:
1335:
1332:
1329:
1324:
1320:
1316:
1311:
1307:
1303:
1298:
1294:
1283:
1280:
1277:
1273:
1262:
1259:Closed under
1258:
1257:
1256:
1254:
1252:
1250:
1249:
1220:
1194:
1191:
1179:
1166:
1163:
1155:
1151:
1123:
1107:
1056:
1053:
1033:
1025:
1021:
976:
966:
961:
957:
953:
950:
947:
942:
938:
929:
926:
923:
908:
903:
899:
895:
892:
889:
884:
880:
869:
868:
854:
844:
839:
835:
831:
828:
825:
820:
816:
807:
804:
801:
786:
781:
777:
773:
770:
767:
762:
758:
747:
746:
732:
722:
719:
716:
713:
698:
695:
692:
689:
680:
679:intersections
676:
675:
661:
651:
648:
645:
642:
627:
624:
621:
618:
609:
605:
604:
603:
601:
585:
575:
572:
564:
563:
550:
540:
537:
515:
508:(or contains
507:
504:Contains the
503:
502:
488:
478:
475:
460:
457:
451:
430:
422:
419:Closed under
418:
417:
416:
415:
413:
400:
397:
390:algebra over
375:
372:
364:
337:
321:
314:
288:
285:
268:
266:
261:
259:
243:
235:
231:
227:
222:
220:
219:intersections
217:, and finite
216:
200:
197:
189:
185:
181:
168:
161:algebra over
146:
138:
111:
95:
88:
62:
59:
48:
44:
43:field of sets
40:
30:
19:
7520:
7447:
7445:
7436:Intersection
6512:
5219:
5201:
5194:
5177:Stone spaces
5176:
5165:
5158:
5133:
5087:Set function
5081:Ring of sets
4996:Boolean ring
4967:
4964:group theory
4955:
4953:
4946:
4941:
4937:
4936:is called a
4853:having both
4848:
4589:and for all
4275:
4271:
4107:
4097:
4088:
4076:
4074:
4069:
3933:
3929:
3927:
3909:
3901:
3899:
3693:
3577:is a triple
3574:
3572:
3561:
3557:
3549:Stone fields
3548:
3546:
3542:
3537:
3535:
3522:
3505:
3488:
3475:
3253:is a triple
3250:
3248:
3228:
3216:sample space
3215:
3211:
3207:
3204:sample space
3203:
3199:
3072:is a triple
3069:
3067:
3062:
3050:
3046:
3029:
3004:
2998:
2765:is always a
2586:
2556:
2549:
2545:
2533:truth tables
2525:
2509:order theory
2500:
2496:ultrafilters
2492:
2480:
2470:
2446:quotient set
2403:
2305:for a point
2199:
1888:with finite
1582:
1246:
1245:is called a
1218:
1217:is called a
1180:
1153:
1149:
1121:
993:
389:
274:
262:
223:
160:
42:
36:
7557:of sets in
7523:semialgebra
7491:of sets in
7336:complements
7329:complements
5866:(Semifield)
5863:Semialgebra
5099:Sigma-ideal
3486:modal logic
3007:Stone space
2999:descriptive
2950:clopen sets
2911:is compact.
1148:are called
1120:are called
271:Definitions
230:ring theory
188:complements
39:mathematics
18:Set algebra
7710:Categories
6670:Dual ideal
6595:(𝜎-Field)
6592:𝜎-Algebra
5149:References
4949:isomorphic
2546:separative
2142:(that is,
1020:subalgebra
388:called an
159:called an
7684:∅
7681:≠
7630:…
7538:∖
7535:Ω
7472:∖
7416:∅
7408:contains
7390:Ω
7382:contains
7373:countable
7364:countable
7346:Ω
7250::
7191:∩
7099:∪
7002:∅
6907:∅
6844:Prefilter
6809:∅
6177:they are
6163:↗
6114:⊆
6071:𝜆-system
6032:↗
6000:↘
5672:∈
5669:∅
5639:∈
5636:Ω
5613:⋯
5610:∪
5597:∪
5564:⋯
5561:∩
5548:∩
5512:∖
5509:Ω
5483:∖
5454:∪
5425:∩
5398:⊇
5337::
5278:Ω
5244:Families
5226:EMS Press
5093:σ-algebra
4855:operators
4808:…
4764:∈
4748:…
4732:∈
4708:∈
4678:…
4626:∈
4610:…
4468:∈
4410:∅
4404:′
4398:∪
4392:∩
4232:relations
4077:canonical
4053:∈
4027:∈
3997:∈
3971:≤
3930:algebraic
3880:≤
3866:∈
3849:∈
3797:≤
3783:∈
3766:∈
3710:∈
3636:≤
3594:≤
3538:algebraic
3501:semantics
3182:μ
3158:μ
3099:μ
3036:countable
2511:based on
2342:∈
2316:∈
2290:∈
2279:whenever
2153:⊂
2120:∈
2090:∈
2061:∈
2046:∣
2040:∈
2014:−
1925:→
1608:power set
1546:∈
1543:…
1490:∈
1487:⋯
1484:∩
1471:∩
1443:∞
1428:⋂
1392:∈
1389:…
1336:∈
1333:⋯
1330:∪
1317:∪
1289:∞
1274:⋃
1261:countable
1248:σ-algebra
1150:complexes
1057:∈
967:∈
951:…
927:≥
909:∈
896:∩
893:⋯
890:∩
845:∈
829:…
805:≥
787:∈
774:∪
771:⋯
768:∪
723:∈
699:∈
693:∩
652:∈
628:∈
622:∪
576:∈
541:∈
538:∅
506:empty set
479:∈
461:∈
455:∖
232:nor with
184:empty set
7450:semiring
7438:Property
7327:relative
7302:downward
7300:directed
7005:∉
6910:∉
6812:∉
6514:Algebra
6179:disjoint
6145:only if
6014:only if
5982:only if
5787:Semiring
5387:Directed
5175:(1982).
4974:See also
4042:implies
3484:for the
3059:Sikorski
2589:topology
2485:complete
1509:for all
1355:for all
1018:forms a
7459:-system
7267:or, is
6516:(Field)
6439:𝜎-Ring
6102:only if
5708:-system
5354:or, is
5268:of sets
5228:, 2001
5105:𝜎-ring
5060:-system
4968:complex
4956:complex
3904:in the
3231:Physics
3172:measure
3015:duality
2557:compact
2517:filters
363:subsets
213:finite
137:subsets
7434:Finite
7375:unions
7320:unions
7318:finite
7309:finite
7228:Never
7129:Never
6749:Filter
6663:Never
6585:Never
6507:Never
6432:Never
6364:δ-Ring
6357:Never
6197:Never
5935:Never
5855:Never
5696:F.I.P.
5183:
5006:δ
4174:where
3619:where
3525:clopen
3507:Every
3495:) and
3299:where
3212:events
3114:where
3055:Loomis
3053:. The
3032:unions
2707:where
2561:filter
2513:ideals
2488:atomic
2448:; and
2079:where
1263:unions
1122:points
608:unions
336:family
334:and a
226:fields
215:unions
110:family
108:and a
7454:is a
6971:Never
6968:Never
6876:Never
6873:Never
6778:Never
6775:Never
6286:Ring
6204:Ring
5270:over
5125:Notes
5008:-ring
4960:group
4901:then
4542:then
4226:is a
3934:tight
3651:is a
3335:is a
3170:is a
2948:with
2944:is a
2861:is a
2802:is a
2563:over
2477:atoms
2222:atoms
1805:with
45:is a
5181:ISBN
4274:(or
3932:(or
3815:and
3655:and
3339:and
3222:and
2638:base
2548:(or
2109:and
1991:via
1964:and
1606:its
41:, a
7338:in
5389:by
4970:.)
4940:or
4486:if
3516:of
3435:on
3379:of
3237:or
2515:or
2244:by
1419::
1265::
1156:of
1026:of
872::
750::
682::
611::
423:in
365:of
361:of
313:set
228:in
190:in
139:of
135:of
87:set
37:In
7712::
7519:A
6175:or
5224:,
5218:,
5157:,
4944:.
4098:S4
4070:S4
4016:,
3910:S4
3729::
3573:A
3520:.
3489:S4
3249:A
3068:A
2535:.
2523:.
2424:;
1995::
1458::=
1304::=
444::
221:.
7687:.
7676:F
7652:F
7627:,
7622:2
7618:A
7614:,
7609:1
7605:A
7601:,
7598:B
7595:,
7592:A
7572:.
7567:F
7541:A
7506:.
7501:F
7475:A
7469:B
7457:π
7277:F
7245:F
7203:)
7111:)
7010:F
6915:F
6817:F
6158:i
6154:A
6117:B
6111:A
6027:i
6023:A
5995:i
5991:A
5706:π
5677:F
5644:F
5605:2
5601:A
5592:1
5588:A
5556:2
5552:A
5543:1
5539:A
5515:A
5486:A
5480:B
5457:B
5451:A
5428:B
5422:A
5364:F
5332:F
5310:e
5303:t
5296:v
5254:F
5189:.
5058:π
4924:)
4920:X
4916:(
4911:C
4889:X
4867:F
4834:}
4830:)
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4631:F
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3883:y
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2186:.
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2150:B
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2067:}
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2058:)
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2010:f
2006:=
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1977:Y
1973:2
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