1059:
with specialization order ≤ is finer than the upper topology and coarser than the Scott topology. Yet, such a space may fail to exist, that is, there exist partial orders for which there is no sober order-consistent topology. Especially, the Scott topology is not necessarily sober.
1017:
The specialization order yields a tool to obtain a preorder from every topology. It is natural to ask for the converse too: Is every preorder obtained as a specialization preorder of some topology?
1009:
with respect to a certain order ≤ if it induces ≤ as its specialization order and it has the above property of inaccessibility with respect to (existing) suprema of directed sets in ≤.
397:, in which closed points are the most specific, while a generic point of a space is one contained in every nonempty open subset. Specialization as an idea is applied also in
884:
1074:
1047:
There are also interesting topologies in between these two extremes. The finest sober topology that is order consistent in the above sense for a given order ≤ is the
837:
807:
1810:
1793:
1323:
1081:, see especially Chapter 5, that explains the motivations from the viewpoint of denotational semantics in computer science. See also the author's
1159:
1028:
of the order ≤ plays a special role: it is the finest topology that induces ≤. The other extreme, the coarsest topology that induces ≤, is the
1640:
1122:
377:
may or may not have. The more closed sets contain a point, the more properties the point has, and the more special it is. The usage is
1776:
1635:
1051:. The upper topology however is still the coarsest sober order-consistent topology. In fact, its open sets are even inaccessible by
1630:
1266:
937:
670:
373:, since it is contained in more open sets. This is particularly intuitive if one views closed sets as properties that a point
1348:
895:
1667:
1587:
96:. The specialization order is also important for identifying suitable topologies on partially ordered sets, as is done in
1261:
1452:
1381:
1355:
1343:
1306:
1281:
1256:
1210:
1179:
899:
1286:
1276:
128:
of the other. However, various authors disagree on which 'direction' the order should go. What is agreed is that if
1652:
1152:
1291:
994:
441:
156:
1557:
1184:
1805:
1788:
591:{0,1} with open sets {∅, {1}, {0,1}} the specialization order is the natural one (0 ≤ 0, 0 ≤ 1, and 1 ≤ 1).
1717:
1333:
698:
93:
1843:
1695:
1530:
1521:
1390:
1225:
1189:
1145:
787:
300:
For the sake of consistency, for the remainder of this article we will take the first definition, that "
1271:
1783:
1742:
1732:
1722:
1467:
1430:
1420:
1400:
1385:
1025:
666:
842:
1710:
1621:
1567:
1526:
1516:
1405:
1338:
1301:
907:
659:
1020:
Indeed, the answer to this question is positive and there are in general many topologies on a set
1749:
1602:
1511:
1501:
1442:
1360:
1102:
810:
729:
655:
607:
394:
270:
125:
1822:
1662:
1296:
1848:
1759:
1737:
1597:
1582:
1562:
1365:
444:
if and only if every upper set is also open (or equivalently every lower set is also closed).
52:
816:
1572:
1425:
1098:
611:
588:
398:
263:
85:
62:
1754:
1537:
1415:
1410:
1395:
1220:
1205:
1118:
887:
780:
571:
1311:
55:. For most spaces that are considered in practice, namely for all those that satisfy the
1078:
1672:
1657:
1647:
1506:
1484:
1462:
1048:
1029:
792:
256:
236:
17:
1837:
1771:
1727:
1705:
1577:
1447:
1435:
1240:
642:
405:
390:
148:
66:
755:
are topologically indistinguishable. It follows that if the underlying topology is T
1592:
1474:
1457:
1375:
1215:
1168:
941:
903:
225:
Both definitions have intuitive justifications: in the case of the former, we have
97:
902:
that assigns a topological space its specialization preorder. This functor has a
1798:
1491:
1370:
1235:
1056:
918:
654:
As suggested by the name, the specialization preorder is a preorder, i.e. it is
634:
28:
1766:
1700:
1541:
433:
378:
332:
160:
1817:
1690:
1496:
991:
437:
429:
408:, a branch of order theory that has ample applications in computer science.
440:. The converses are not generally true. In fact, a topological space is an
404:
The intuition of upper elements being more specific is typically found in
1612:
1479:
1230:
952:
733:
425:
365:
These restatements help to explain why one speaks of a "specialization":
354:
74:
56:
48:
32:
891:
386:
566:
need not be open or closed. The closed points of a topological space
420:
be a topological space and let ≤ be the specialization preorder on
1001:
One may describe the second property by saying that open sets are
886:
The converse, however, is not true in general. In the language of
382:
1082:
921:. Their relationship to the specialization order is more subtle:
269:(which is the motivational situation in applications related to
84:
The specialization order is often considered in applications in
1141:
1137:
1024:
that induce a given order ≤ as their specialization order. The
813:
with respect to the specialization preorders of these spaces:
775:. Hence, the specialization order is of little interest for T
1077:, 1998. Revised version of author's Ph.D. thesis. Available
1032:, the least topology within which all complements of sets ↓
759:, then the specialization order is discrete, i.e. one has
669:
determined by the specialization preorder is just that of
273:), then under our second definition of the order, we have
1130:, vol. 142, Trans. Amer. Math. Soc., pp. 43–60
845:
819:
795:
81:
the order becomes trivial and is of little interest.
732:
of the specialization preorder is equivalent to the
681:are topologically indistinguishable if and only if
1683:
1611:
1550:
1320:
1249:
1198:
878:
831:
801:
222:" by various authors (see, respectively, and ).
1075:Electronic Notes in Theoretical Computer Science
917:spaces for which this order is interesting: the
721:. In this case it is justified to speak of the
913:There are spaces that are more specific than T
1153:
8:
1811:Positive cone of a partially ordered group
1160:
1146:
1138:
1124:Prime ideal structure in commutative rings
562:is always closed; however, the upper set ↑
844:
818:
794:
1794:Positive cone of an ordered vector space
479:} is a singleton one uses the notation ↑
1090:
389:; and also with the traditional use of
1107:, New York-Heidelberg: Springer-Verlag
463:and the smallest lower set containing
381:with the classical logical notions of
928:with specialization order ≤, we have
251:However, in the case where our space
7:
455:. The smallest upper set containing
1321:Properties & Types (
809:between two topological spaces is
637:). Thus the closed points of Spec(
25:
1777:Positive cone of an ordered field
1631:Ordered topological vector space
1071:Topological Duality in Semantics
1003:inaccessible by directed suprema
671:topological indistinguishability
938:directed complete partial order
779:topologies, especially for all
206:" is alternatively written as "
108:Consider any topological space
896:category of topological spaces
879:{\displaystyle f(x)\leq f(y).}
870:
864:
855:
849:
51:on the set of the points of a
1:
1588:Series-parallel partial order
546:} = ∩{closed sets containing
198:Unfortunately, the property "
147:} denotes the closure of the
1267:Cantor's isomorphism theorem
432:with respect to ≤ and every
292:as prime ideals of the ring
1307:Szpilrajn extension theorem
1282:Hausdorff maximal principle
1257:Boolean prime ideal theorem
900:category of preordered sets
713:are indistinguishable then
519:} = ∩{open sets containing
191:; this is commonly written
1865:
1653:Topological vector lattice
962:for every directed subset
73:). On the other hand, for
65:, this preorder is even a
1175:
442:Alexandrov-discrete space
104:Definition and motivation
1262:Cantor–Bernstein theorem
970:, ≤) and every open set
1806:Partially ordered group
1626:Specialization preorder
832:{\displaystyle x\leq y}
728:On the other hand, the
701:of ≤ is precisely the T
304:is a specialization of
202:is a specialization of
114:specialization preorder
18:Specialization preorder
1292:Kruskal's tree theorem
1287:Knaster–Tarski theorem
1277:Dushnik–Miller theorem
880:
833:
803:
120:relates two points of
94:denotational semantics
910:on a preordered set.
881:
834:
804:
705:separation axiom: if
602:are elements of Spec(
369:is more general than
124:when one lies in the
1784:Ordered vector space
1026:Alexandroff topology
1013:Topologies on orders
924:For any sober space
843:
817:
793:
723:specialization order
667:equivalence relation
650:Important properties
641:) are precisely the
412:Upper and lower sets
353:is contained in all
331:is contained in all
71:specialization order
1622:Alexandrov topology
1568:Lexicographic order
1527:Well-quasi-ordering
1055:suprema. Hence any
908:Alexandrov topology
906:, which places the
788:continuous function
578:with respect to ≤.
135:is contained in cl{
1603:Transitive closure
1563:Converse/Transpose
1272:Dilworth's theorem
1104:Algebraic geometry
876:
829:
799:
739:separation axiom:
570:are precisely the
395:algebraic geometry
271:algebraic geometry
1831:
1830:
1789:Partially ordered
1598:Symmetric closure
1583:Reflexive closure
1326:
1099:Hartshorne, Robin
890:, we then have a
802:{\displaystyle f}
697:. Therefore, the
53:topological space
27:In the branch of
16:(Redirected from
1856:
1573:Linear extension
1322:
1302:Mirsky's theorem
1162:
1155:
1148:
1139:
1132:
1131:
1129:
1119:Hochster, Melvin
1115:
1109:
1108:
1095:
1069:M.M. Bonsangue,
1007:order consistent
1005:. A topology is
885:
883:
882:
877:
838:
836:
835:
830:
808:
806:
805:
800:
781:Hausdorff spaces
612:commutative ring
589:Sierpinski space
572:minimal elements
399:valuation theory
308:" be written as
264:commutative ring
167:}), we say that
92:spaces occur in
86:computer science
63:separation axiom
21:
1864:
1863:
1859:
1858:
1857:
1855:
1854:
1853:
1834:
1833:
1832:
1827:
1823:Young's lattice
1679:
1607:
1546:
1396:Heyting algebra
1344:Boolean algebra
1316:
1297:Laver's theorem
1245:
1211:Boolean algebra
1206:Binary relation
1194:
1171:
1166:
1136:
1135:
1127:
1117:
1116:
1112:
1097:
1096:
1092:
1066:
1015:
942:directed subset
916:
888:category theory
841:
840:
815:
814:
791:
790:
778:
767:if and only if
758:
747:if and only if
737:
704:
652:
625:if and only if
584:
558:The lower set ↓
451:be a subset of
414:
349:if and only if
327:if and only if
316:. We then see,
284:if and only if
106:
91:
78:
60:
23:
22:
15:
12:
11:
5:
1862:
1860:
1852:
1851:
1846:
1836:
1835:
1829:
1828:
1826:
1825:
1820:
1815:
1814:
1813:
1803:
1802:
1801:
1796:
1791:
1781:
1780:
1779:
1769:
1764:
1763:
1762:
1757:
1750:Order morphism
1747:
1746:
1745:
1735:
1730:
1725:
1720:
1715:
1714:
1713:
1703:
1698:
1693:
1687:
1685:
1681:
1680:
1678:
1677:
1676:
1675:
1670:
1668:Locally convex
1665:
1660:
1650:
1648:Order topology
1645:
1644:
1643:
1641:Order topology
1638:
1628:
1618:
1616:
1609:
1608:
1606:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1565:
1560:
1554:
1552:
1548:
1547:
1545:
1544:
1534:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1488:
1487:
1477:
1472:
1471:
1470:
1465:
1460:
1455:
1453:Chain-complete
1445:
1440:
1439:
1438:
1433:
1428:
1423:
1418:
1408:
1403:
1398:
1393:
1388:
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1373:
1368:
1363:
1358:
1353:
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1351:
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1200:
1196:
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1187:
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1167:
1165:
1164:
1157:
1150:
1142:
1134:
1133:
1110:
1089:
1088:
1087:
1086:
1073:, volume 8 of
1065:
1062:
1049:Scott topology
1030:upper topology
1014:
1011:
999:
998:
960:
914:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
828:
825:
822:
798:
776:
756:
735:
702:
651:
648:
647:
646:
643:maximal ideals
592:
583:
580:
556:
555:
524:
413:
410:
391:generic points
363:
362:
340:
298:
297:
257:prime spectrum
249:
248:
237:if and only if
185:generalization
173:specialization
141:
140:
105:
102:
89:
76:
58:
37:specialization
24:
14:
13:
10:
9:
6:
4:
3:
2:
1861:
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1821:
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1816:
1812:
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1800:
1797:
1795:
1792:
1790:
1787:
1786:
1785:
1782:
1778:
1775:
1774:
1773:
1772:Ordered field
1770:
1768:
1765:
1761:
1758:
1756:
1753:
1752:
1751:
1748:
1744:
1741:
1740:
1739:
1736:
1734:
1731:
1729:
1728:Hasse diagram
1726:
1724:
1721:
1719:
1716:
1712:
1709:
1708:
1707:
1706:Comparability
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1688:
1686:
1682:
1674:
1671:
1669:
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1659:
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1655:
1654:
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1649:
1646:
1642:
1639:
1637:
1634:
1633:
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1629:
1627:
1623:
1620:
1619:
1617:
1614:
1610:
1604:
1601:
1599:
1596:
1594:
1591:
1589:
1586:
1584:
1581:
1579:
1578:Product order
1576:
1574:
1571:
1569:
1566:
1564:
1561:
1559:
1556:
1555:
1553:
1551:Constructions
1549:
1543:
1539:
1535:
1532:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
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1500:
1498:
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1490:
1486:
1483:
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1481:
1478:
1476:
1473:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1450:
1449:
1448:Partial order
1446:
1444:
1441:
1437:
1436:Join and meet
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1413:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
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1392:
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1383:
1379:
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1369:
1367:
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1362:
1359:
1357:
1354:
1350:
1347:
1346:
1345:
1342:
1340:
1337:
1335:
1334:Antisymmetric
1332:
1331:
1329:
1325:
1319:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1295:
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1290:
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1283:
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1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1254:
1252:
1248:
1242:
1241:Weak ordering
1239:
1237:
1234:
1232:
1229:
1227:
1226:Partial order
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1203:
1201:
1197:
1191:
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1125:
1120:
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1111:
1106:
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1100:
1094:
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1084:
1080:
1076:
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1039:
1035:
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1027:
1023:
1018:
1012:
1010:
1008:
1004:
996:
993:
989:
985:
981:
977:
973:
969:
965:
961:
958:
954:
950:
946:
943:
940:, i.e. every
939:
935:
931:
930:
929:
927:
922:
920:
911:
909:
905:
901:
897:
893:
889:
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823:
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812:
796:
789:
784:
782:
774:
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750:
746:
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409:
407:
406:domain theory
402:
400:
396:
392:
388:
384:
380:
376:
372:
368:
360:
357:that contain
356:
352:
348:
344:
341:
338:
335:that contain
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303:
295:
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287:
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196:
194:
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186:
182:
178:
174:
170:
166:
163:containing {
162:
158:
154:
150:
149:singleton set
146:
138:
134:
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129:
127:
123:
119:
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111:
103:
101:
99:
95:
87:
82:
80:
72:
68:
67:partial order
64:
61:
54:
50:
47:is a natural
46:
42:
38:
34:
30:
19:
1844:Order theory
1625:
1615:& Orders
1593:Star product
1522:Well-founded
1475:Prefix order
1431:Distributive
1421:Complemented
1391:Foundational
1356:Completeness
1312:Zorn's lemma
1216:Cyclic order
1199:Key concepts
1169:Order theory
1123:
1113:
1103:
1093:
1070:
1052:
1046:
1044:) are open.
1041:
1037:
1033:
1021:
1019:
1016:
1006:
1002:
1000:
995:intersection
987:
983:
979:
975:
971:
967:
963:
956:
948:
944:
933:
925:
923:
919:sober spaces
912:
904:left adjoint
785:
772:
768:
764:
760:
752:
748:
744:
740:
727:
722:
718:
714:
710:
706:
699:antisymmetry
694:
690:
686:
682:
678:
674:
664:
653:
638:
635:prime ideals
630:
626:
622:
618:
614:
603:
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467:is denoted ↓
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155:}, i.e. the
152:
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113:
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98:order theory
83:
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69:(called the
44:
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1799:Riesz space
1760:Isomorphism
1636:Normal cone
1558:Composition
1492:Semilattice
1401:Homogeneous
1386:Equivalence
1236:Total order
1057:sober space
951:, ≤) has a
673:. That is,
333:closed sets
161:closed sets
29:mathematics
1838:Categories
1767:Order type
1701:Cofinality
1542:Well-order
1517:Transitive
1406:Idempotent
1339:Asymmetric
1064:References
1036:(for some
936:, ≤) is a
660:transitive
471:. In case
434:closed set
379:consistent
214:" and as "
143:(where cl{
1818:Upper set
1755:Embedding
1691:Antichain
1512:Tolerance
1502:Symmetric
1497:Semiorder
1443:Reflexive
1361:Connected
992:non-empty
974:, if sup
894:from the
859:≤
824:≤
656:reflexive
495:one has:
438:lower set
430:upper set
355:open sets
179:and that
88:, where T
41:canonical
31:known as
1849:Topology
1613:Topology
1480:Preorder
1463:Eulerian
1426:Complete
1376:Directed
1366:Covering
1231:Preorder
1190:Category
1185:Glossary
1121:(1969),
1101:(1977),
1083:homepage
953:supremum
839:implies
811:monotone
730:symmetry
608:spectrum
582:Examples
538: :
511: :
426:open set
424:. Every
49:preorder
45:preorder
33:topology
1718:Duality
1696:Cofinal
1684:Related
1663:Fréchet
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1416:Bounded
1411:Lattice
1384:)
1382:Partial
1250:Results
1221:Lattice
982:, then
898:to the
892:functor
617:) then
606:) (the
587:In the
550:} = cl{
387:species
255:is the
243:} ⊆ cl{
159:of all
126:closure
1743:Subnet
1723:Filter
1673:Normed
1658:Banach
1624:&
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1468:Strict
1458:Graded
1349:topics
1180:Topics
1079:online
978:is in
487:. For
428:is an
260:Spec R
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79:spaces
35:, the
1733:Ideal
1711:Graph
1507:Total
1485:Total
1371:Dense
1128:(PDF)
990:have
689:and
610:of a
483:and ↓
436:is a
383:genus
262:of a
193:y ⤳ x
183:is a
171:is a
116:≤ on
1324:list
986:and
966:of (
955:sup
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786:Any
751:and
709:and
677:and
665:The
658:and
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416:Let
385:and
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