2286:
and preorders. C. Naturman observed that these spaces were the
Alexandrov-discrete spaces and extended the result to a category-theoretic equivalence between the category of Alexandrov-discrete spaces and (open) continuous maps, and the category of preorders and (bounded) monotone maps, providing the
1486:
Notice however that in the case of topologies other than the
Alexandrov topology, we can have a map between two topological spaces that is not continuous but which is nevertheless still a monotone function between the corresponding preordered sets. (To see this consider a non-Alexandrov-discrete
1482:
Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding
Alexandrov-discrete spaces. Conversely a map between two Alexandrov-discrete spaces is continuous if and only if it is a monotone function between the corresponding preordered sets.
814:
994:
868:
2178:
later came to be used for topological spaces in which every subset is open and the original concept lay forgotten in the topological literature. On the other hand, Alexandrov spaces played a relevant role in
663:
2294:
A systematic investigation of these spaces from the point of view of general topology, which had been neglected since the original paper by
Alexandrov was taken up by F. G. Arenas.
699:
885:
2486:
2149:
683:
2123:
2103:
2083:
2063:
2043:
2023:
2003:
3269:
3312:
2253:
3252:
2782:
2618:
2540:
3099:
822:
2244:. F. G. Arenas independently proposed this name for the general version of these topologies. McCord also showed that these spaces are
1952:
i.e. a set with relations defined on it.) The class of modal algebras that we obtain in the case of a preordered set is the class of
3235:
3094:
3089:
138:
2210:
was adopted for them. Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from
2725:
2807:
1549:
3126:
3046:
628:
2720:
2911:
2840:
1945:
2814:
2802:
2765:
2740:
2715:
2669:
2638:
1925:
2745:
2735:
3302:
3111:
2611:
3084:
2750:
2453:
2245:
2199:
1396:
123:
93:
40:
3016:
2643:
1682:
213:
108:
3264:
3247:
1021:
292:
66:
809:{\displaystyle \tau =\{\,G\subseteq X:\forall x,y\in X\ \ (x\in G\ \land \ x\leq y)\ \implies \ y\in G\,\}}
3176:
2792:
2211:
2155:
107:
the family of all finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of
3307:
3154:
2989:
2980:
2849:
2684:
2648:
2604:
2226:
1553:
103:
Alexandrov-discrete spaces are also called finitely generated spaces because their topology is uniquely
2730:
1985:
of compact neighbourhoods, since the smallest neighbourhood of a point is always compact. Indeed, if
1668:
1605:
3242:
3201:
3191:
3181:
2926:
2889:
2879:
2859:
2844:
2560:
2270:
that a equivalence exists between finite topological spaces and preorders on finite sets (the finite
2195:
1949:
989:{\displaystyle \{\,S\subseteq X:\forall x,y\in X\ \ (x\in S\ \land \ y\leq x)\ \implies \ y\in S\,\}}
223:
3169:
3026:
2985:
2975:
2864:
2797:
2760:
2279:
246:
119:
115:
3208:
3061:
2970:
2960:
2901:
2819:
2513:
2430:
2352:
262:
3281:
3121:
2755:
547:
532:
2225:
In 1966 Michael C. McCord and A. K. Steiner each independently observed an equivalence between
3218:
3196:
3056:
3041:
3021:
2824:
2536:
2505:
1831:
1565:
1541:
1494:
1262:
484:
104:
74:
36:
3031:
2884:
2495:
2462:
2451:
McCord, M. C. (1966). "Singular homology and homotopy groups of finite topological spaces".
2309:, a space satisfying the weaker condition that countable intersections of open sets are open
2288:
2260:
2252:
of the corresponding partially ordered set. Steiner demonstrated that the equivalence is a
2230:
2219:
2203:
2184:
2167:
1953:
1937:
1835:
1624:
1613:
1537:
134:
130:
2128:
668:
3213:
2996:
2874:
2869:
2854:
2679:
2664:
2256:
2237:
1978:
1971:
1941:
288:
2770:
122:
and intersections, the property of being an
Alexandrov-discrete space is preserved under
17:
2174:, where he provided the characterizations in terms of sets and neighbourhoods. The name
1944:. (The latter construction is itself a special case of a more general construction of a
3131:
3116:
3106:
2965:
2943:
2921:
2188:
2180:
2175:
2108:
2088:
2068:
2048:
2028:
2008:
1988:
623:
51:
family of open sets is open; in
Alexandrov topologies the finite qualifier is dropped.
3296:
3230:
3186:
3164:
3036:
2906:
2894:
2699:
2249:
2215:
1933:
1921:
Considering the interior operator and closure operator to be modal operators on the
3051:
2933:
2916:
2834:
2674:
2627:
2373:
2466:
1243:)) induces the same specialization preorder as the original topology of the space
2441:, Contemporary mathematics vol. 486, American Mathematical Society, 2009, p.170ff
3257:
2950:
2829:
2694:
2271:
2267:
593:
Topological spaces satisfying the above equivalent characterizations are called
3225:
3159:
3000:
1982:
387:
is in the set. (This preorder will be precisely the specialization preorder.)
359:
is in the set. (This preorder will be precisely the specialization preorder.)
2509:
3276:
3149:
2955:
2404:
2282:
observed that this extended to a equivalence between what he referred to as
1922:
874:
690:
340:
85:
3071:
2938:
2689:
1561:
302:
70:
44:
28:
2351:
Speer, Timothy (16 August 2007). "A Short Study of
Alexandroff Spaces".
2517:
2303:
1964:
Every subspace of an
Alexandrov-discrete space is Alexandrov-discrete.
84:
for which the specialization preorder is ≤. The open sets are just the
2198:
in the 1980s, Alexandrov spaces were rediscovered when the concept of
1967:
The product of two
Alexandrov-discrete spaces is Alexandrov-discrete.
1385:
considered as a map between the corresponding
Alexandrov spaces. Then
2374:"Are minimal neighborhoods in an Alexandrov topology path-connected?"
1166:
is recovered as the topology induced by the specialization preorder.
2500:
2481:
1820:
Relationship to the construction of modal algebras from modal frames
1631:
consisting of the Alexandrov-discrete spaces. Then the restrictions
1469:
considered as a map between the corresponding preordered sets. Then
47:
is open. It is an axiom of topology that the intersection of every
2589:. Ph.D. thesis, University of Cape Town Department of Mathematics.
2357:
863:{\displaystyle \mathbf {T} (\mathbf {X} )=\langle X,\tau \rangle }
129:
Alexandrov-discrete spaces are named after the Russian topologist
2535:(1st paperback ed.). New York: Cambridge University Press.
163:> be a topological space. Then the following are equivalent:
2600:
2596:
1932:, this construction is a special case of the construction of a
474:. (This finite subset can always be chosen to be a singleton.)
550:
of the finite spaces. (This means that there is a final sink
437:
Finite generation and category theoretic characterizations:
149:
Alexandrov topologies have numerous characterizations. Let
2482:"The Lattice of Topologies: Structure and Complementation"
54:
A set together with an Alexandrov topology is known as an
1956:—the algebraic abstractions of topological spaces.
133:. They should not be confused with the more geometrical
2236:
versions of the spaces that Alexandrov had introduced.
1071:
Equivalence between preorders and Alexandrov topologies
65:
Alexandrov topologies are uniquely determined by their
2131:
2111:
2091:
2071:
2051:
2031:
2011:
1991:
1127:) as the specialization preorder. Moreover for every
888:
825:
702:
671:
631:
371:
are precisely those that are downward closed i.e. if
2045:
itself with the subspace topology any open cover of
1225:(i.e. it will have more open sets). The topology of
658:{\displaystyle \mathbf {X} =\langle X,\leq \rangle }
255:
distributes over arbitrary intersections of subsets.
3142:
3070:
3009:
2779:
2708:
2657:
2166:Alexandrov spaces were first introduced in 1937 by
365:There is a preorder ≤ such that the closed sets of
2143:
2117:
2097:
2077:
2057:
2037:
2017:
1997:
1002:The specialization preorder on a topological space
988:
862:
808:
677:
657:
88:with respect to ≤. Thus, Alexandrov topologies on
2487:Transactions of the American Mathematical Society
1777:is an Alexandrov-discrete space, the composition
1528:Category theoretic description of the equivalence
1169:However for a topological space in general we do
333:There is a preorder ≤ such that the open sets of
238:Interior and closure algebraic characterizations:
2437:, in Frédéric Mynard, Elliott Pearl (Editors),
2266:It was also a well-known result in the field of
2105:. Such a neighbourhood is necessarily equal to
2005:is the smallest (open) neighbourhood of a point
1257:Equivalence between monotonicity and continuity
2612:
1286:between two preordered sets (i.e. a function
271:distributes over arbitrary unions of subsets.
8:
2138:
2132:
1709:. This means that given a topological space
983:
889:
857:
845:
803:
709:
652:
640:
423:where ≤ is the specialization preorder i.e.
1749:is continuous and for every continuous map
618:The Alexandrov topology on a preordered set
80:, there is a unique Alexandrov topology on
3270:Positive cone of a partially ordered group
2619:
2605:
2597:
2287:preorder characterizations as well as the
969:
965:
789:
785:
175:An arbitrary intersection of open sets in
145:Characterizations of Alexandrov topologies
2499:
2356:
2130:
2110:
2090:
2070:
2050:
2030:
2010:
1990:
982:
892:
887:
834:
826:
824:
802:
712:
701:
670:
632:
630:
3253:Positive cone of an ordered vector space
2554:
2552:
2429:, Duke Math. J. 10 (1943), 761–785. See
1115:is recovered from the topological space
546:is finitely generated i.e. it is in the
458:if and only if there is a finite subset
232:is closed under arbitrary intersections.
137:introduced by the Russian mathematician
2319:
2390:
2338:
1304:between the underlying sets such that
1249:and is in fact the finest topology on
873:The corresponding closed sets are the
168:Open and closed set characterizations:
2326:
1895: ∈ X : there exists a
665:we can define an Alexandrov topology
187:An arbitrary union of closed sets in
7:
1981:in the sense that every point has a
1424:between two topological spaces, let
689:by choosing the open sets to be the
448:lies within the closure of a subset
301:i.e. the finest topology giving the
2229:and spaces that were precisely the
1219:with a finer topology than that of
819:We thus obtain a topological space
2780:Properties & Types (
1402:Conversely given a continuous map
1160:, i.e. the Alexandrov topology of
905:
725:
25:
3236:Positive cone of an ordered field
2427:Some studies on closure relations
3313:Properties of topological spaces
3090:Ordered topological vector space
1051:We thus obtain a preordered set
835:
827:
633:
613:Equivalence with preordered sets
407:if and only if there is a point
397:lies in the closure of a subset
199:Neighbourhood characterizations:
139:Aleksandr Danilovich Aleksandrov
2240:referred to such topologies as
587:is a finite topological space.)
2587:Interior Algebras and Topology
2289:interior and closure algebraic
1940:i.e. from a set with a single
1865: ∈ S : for all
1550:category of topological spaces
966:
959:
929:
839:
831:
786:
779:
749:
1:
3047:Series-parallel partial order
2467:10.1215/S0012-7094-66-03352-7
2187:and their relationships with
2154:Every Alexandrov topology is
1977:Every Alexandrov topology is
1970:Every Alexandrov topology is
487:with the finite subspaces of
339:are precisely those that are
2726:Cantor's isomorphism theorem
2263:as well as complementation.
2065:contains a neighbourhood of
2766:Szpilrajn extension theorem
2741:Hausdorff maximal principle
2716:Boolean prime ideal theorem
2568:Acta Math. Univ. Comenianae
2284:totally distributive spaces
2125:, so the open cover admits
1683:bico-reflective subcategory
525:of the finite subspaces of
277:Preorder characterizations:
3329:
3112:Topological vector lattice
2411:. New Series (in German).
2378:Mathematics Stack Exchange
1479:) is a monotone function.
1006:Given a topological space
599:Alexandrov-discrete spaces
2634:
2531:Johnstone, P. T. (1986).
2454:Duke Mathematical Journal
2261:arbitrary meets and joins
2208:finitely generated spaces
1129:Alexandrov-discrete space
1075:For every preordered set
595:finitely generated spaces
109:finite topological spaces
94:one-to-one correspondence
56:Alexandrov-discrete space
18:Alexandrov-discrete space
2721:Cantor–Bernstein theorem
2585:Naturman, C. A. (1991).
2403:Alexandroff, P. (1937).
2246:weak homotopy equivalent
2194:With the advancement of
427:lies in the closure of {
282:Specialization preorder.
67:specialization preorders
60:finitely generated space
3265:Partially ordered group
3085:Specialization preorder
2480:Steiner, A. K. (1966).
2259:isomorphism preserving
1824:Given a preordered set
1614:left and right adjoints
1560:denote the category of
1109:, i.e. the preorder of
1085:, ≤> we always have
1022:specialization preorder
470:lies in the closure of
293:specialization preorder
204:Smallest neighbourhood.
118:commute with arbitrary
2751:Kruskal's tree theorem
2746:Knaster–Tarski theorem
2736:Dushnik–Miller theorem
2559:Arenas, F. G. (1999).
2227:partially ordered sets
2212:denotational semantics
2183:pioneering studies on
2156:locally path connected
2151:as a finite subcover.
2145:
2119:
2099:
2079:
2059:
2039:
2019:
1999:
1043:is in the closure of {
990:
864:
810:
679:
659:
317:is in the closure of {
2242:Alexandrov topologies
2146:
2144:{\displaystyle \{U\}}
2120:
2100:
2080:
2060:
2040:
2020:
2000:
1881: ∈ S }, and
1669:concrete isomorphisms
991:
865:
811:
680:
678:{\displaystyle \tau }
660:
497:Finite inclusion map.
220:Neighbourhood filter.
114:Due to the fact that
3243:Ordered vector space
2561:"Alexandroff spaces"
2274:for the modal logic
2196:categorical topology
2129:
2109:
2089:
2069:
2049:
2029:
2009:
1989:
1950:relational structure
1899: ∈ S with
1689:with bico-reflector
1253:with that property.
886:
823:
700:
669:
629:
291:consistent with the
224:neighbourhood filter
69:. Indeed, given any
3081:Alexandrov topology
3027:Lexicographic order
2986:Well-quasi-ordering
2291:characterizations.
1715:, the identity map
1465:be the same map as
1381:be the same map as
1215:)) will be the set
607:Alexandrov topology
601:and their topology
499:The inclusion maps
43:of every family of
33:Alexandrov topology
3062:Transitive closure
3022:Converse/Transpose
2731:Dilworth's theorem
2141:
2115:
2095:
2075:
2055:
2035:
2015:
1995:
1566:monotone functions
986:
860:
806:
675:
655:
539:Finite generation.
375:is in the set and
347:is in the set and
243:Interior operator.
226:of every point in
96:with preorders on
3303:Closure operators
3290:
3289:
3248:Partially ordered
3057:Symmetric closure
3042:Reflexive closure
2785:
2542:978-0-521-33779-3
2200:finite generation
2118:{\displaystyle U}
2098:{\displaystyle U}
2078:{\displaystyle x}
2058:{\displaystyle U}
2038:{\displaystyle U}
2018:{\displaystyle x}
1998:{\displaystyle U}
1954:interior algebras
1832:interior operator
1606:concrete functors
1493:and consider the
1263:monotone function
972:
964:
949:
943:
928:
925:
792:
784:
769:
763:
748:
745:
391:Downward closure.
259:Closure operator.
247:interior operator
135:Alexandrov spaces
16:(Redirected from
3320:
3032:Linear extension
2781:
2761:Mirsky's theorem
2621:
2614:
2607:
2598:
2591:
2590:
2582:
2576:
2575:
2565:
2556:
2547:
2546:
2528:
2522:
2521:
2503:
2477:
2471:
2470:
2448:
2442:
2423:
2417:
2416:
2405:"Diskrete Räume"
2400:
2394:
2388:
2382:
2381:
2370:
2364:
2362:
2360:
2348:
2342:
2336:
2330:
2324:
2220:computer science
2204:general topology
2168:P. S. Alexandrov
2150:
2148:
2147:
2142:
2124:
2122:
2121:
2116:
2104:
2102:
2101:
2096:
2084:
2082:
2081:
2076:
2064:
2062:
2061:
2056:
2044:
2042:
2041:
2036:
2024:
2022:
2021:
2016:
2004:
2002:
2001:
1996:
1869: ∈ X,
1850:) are given by:
1836:closure operator
1625:full subcategory
1538:category of sets
995:
993:
992:
987:
970:
962:
947:
941:
926:
923:
869:
867:
866:
861:
838:
830:
815:
813:
812:
807:
790:
782:
767:
761:
746:
743:
684:
682:
681:
676:
664:
662:
661:
656:
636:
478:Finite subspace.
363:Closed down-set.
263:closure operator
131:Pavel Alexandrov
21:
3328:
3327:
3323:
3322:
3321:
3319:
3318:
3317:
3293:
3292:
3291:
3286:
3282:Young's lattice
3138:
3066:
3005:
2855:Heyting algebra
2803:Boolean algebra
2775:
2756:Laver's theorem
2704:
2670:Boolean algebra
2665:Binary relation
2653:
2630:
2625:
2595:
2594:
2584:
2583:
2579:
2563:
2558:
2557:
2550:
2543:
2530:
2529:
2525:
2501:10.2307/1994555
2479:
2478:
2474:
2450:
2449:
2445:
2439:Beyond Topology
2424:
2420:
2402:
2401:
2397:
2389:
2385:
2372:
2371:
2367:
2350:
2349:
2345:
2337:
2333:
2325:
2321:
2316:
2300:
2238:P. T. Johnstone
2234:
2202:was applied to
2185:closure systems
2176:discrete spaces
2172:discrete spaces
2170:under the name
2164:
2127:
2126:
2107:
2106:
2087:
2086:
2067:
2066:
2047:
2046:
2027:
2026:
2007:
2006:
1987:
1986:
1979:locally compact
1972:first countable
1962:
1946:complex algebra
1942:binary relation
1926:Boolean algebra
1822:
1816:is continuous.
1562:preordered sets
1554:continuous maps
1530:
1259:
1073:
1039:if and only if
1028:is defined by:
1004:
884:
883:
821:
820:
698:
697:
667:
666:
627:
626:
620:
615:
586:
569:
558:
518:
507:
442:Finite closure.
313:if and only if
289:finest topology
212:has a smallest
206:Every point of
147:
23:
22:
15:
12:
11:
5:
3326:
3324:
3316:
3315:
3310:
3305:
3295:
3294:
3288:
3287:
3285:
3284:
3279:
3274:
3273:
3272:
3262:
3261:
3260:
3255:
3250:
3240:
3239:
3238:
3228:
3223:
3222:
3221:
3216:
3209:Order morphism
3206:
3205:
3204:
3194:
3189:
3184:
3179:
3174:
3173:
3172:
3162:
3157:
3152:
3146:
3144:
3140:
3139:
3137:
3136:
3135:
3134:
3129:
3127:Locally convex
3124:
3119:
3109:
3107:Order topology
3104:
3103:
3102:
3100:Order topology
3097:
3087:
3077:
3075:
3068:
3067:
3065:
3064:
3059:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3019:
3013:
3011:
3007:
3006:
3004:
3003:
2993:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2948:
2947:
2946:
2936:
2931:
2930:
2929:
2924:
2919:
2914:
2912:Chain-complete
2904:
2899:
2898:
2897:
2892:
2887:
2882:
2877:
2867:
2862:
2857:
2852:
2847:
2837:
2832:
2827:
2822:
2817:
2812:
2811:
2810:
2800:
2795:
2789:
2787:
2777:
2776:
2774:
2773:
2768:
2763:
2758:
2753:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2712:
2710:
2706:
2705:
2703:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2661:
2659:
2655:
2654:
2652:
2651:
2646:
2641:
2635:
2632:
2631:
2626:
2624:
2623:
2616:
2609:
2601:
2593:
2592:
2577:
2548:
2541:
2523:
2494:(2): 379–398.
2472:
2461:(3): 465–474.
2443:
2418:
2395:
2393:, Theorem 2.8.
2383:
2365:
2343:
2341:, Theorem 2.2.
2331:
2318:
2317:
2315:
2312:
2311:
2310:
2299:
2296:
2280:A. Grzegorczyk
2232:
2191:and topology.
2189:lattice theory
2163:
2160:
2140:
2137:
2134:
2114:
2094:
2074:
2054:
2034:
2014:
1994:
1961:
1958:
1909:
1908:
1882:
1821:
1818:
1814:
1813:
1769:
1768:
1747:
1746:
1665:
1664:
1649:
1616:respectively.
1602:
1601:
1586:
1529:
1526:
1463:
1462:
1437:) :
1422:
1421:
1397:continuous map
1379:
1378:
1353:) :
1326:) ≤
1302:
1301:
1284:
1283:
1258:
1255:
1072:
1069:
1049:
1048:
1003:
1000:
999:
998:
997:
996:
985:
981:
978:
975:
968:
961:
958:
955:
952:
946:
940:
937:
934:
931:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
891:
859:
856:
853:
850:
847:
844:
841:
837:
833:
829:
817:
816:
805:
801:
798:
795:
788:
781:
778:
775:
772:
766:
760:
757:
754:
751:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
711:
708:
705:
674:
654:
651:
648:
645:
642:
639:
635:
624:preordered set
619:
616:
614:
611:
591:
590:
589:
588:
582:
565:
554:
536:
514:
503:
494:
475:
434:
433:
432:
388:
360:
328:
274:
273:
272:
256:
235:
234:
233:
217:
196:
195:
194:
182:
146:
143:
116:inverse images
24:
14:
13:
10:
9:
6:
4:
3:
2:
3325:
3314:
3311:
3309:
3306:
3304:
3301:
3300:
3298:
3283:
3280:
3278:
3275:
3271:
3268:
3267:
3266:
3263:
3259:
3256:
3254:
3251:
3249:
3246:
3245:
3244:
3241:
3237:
3234:
3233:
3232:
3231:Ordered field
3229:
3227:
3224:
3220:
3217:
3215:
3212:
3211:
3210:
3207:
3203:
3200:
3199:
3198:
3195:
3193:
3190:
3188:
3187:Hasse diagram
3185:
3183:
3180:
3178:
3175:
3171:
3168:
3167:
3166:
3165:Comparability
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3147:
3145:
3141:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3114:
3113:
3110:
3108:
3105:
3101:
3098:
3096:
3093:
3092:
3091:
3088:
3086:
3082:
3079:
3078:
3076:
3073:
3069:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3037:Product order
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3014:
3012:
3010:Constructions
3008:
3002:
2998:
2994:
2991:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2945:
2942:
2941:
2940:
2937:
2935:
2932:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2909:
2908:
2907:Partial order
2905:
2903:
2900:
2896:
2895:Join and meet
2893:
2891:
2888:
2886:
2883:
2881:
2878:
2876:
2873:
2872:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2846:
2842:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2809:
2806:
2805:
2804:
2801:
2799:
2796:
2794:
2793:Antisymmetric
2791:
2790:
2788:
2784:
2778:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2713:
2711:
2707:
2701:
2700:Weak ordering
2698:
2696:
2693:
2691:
2688:
2686:
2685:Partial order
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2662:
2660:
2656:
2650:
2647:
2645:
2642:
2640:
2637:
2636:
2633:
2629:
2622:
2617:
2615:
2610:
2608:
2603:
2602:
2599:
2588:
2581:
2578:
2573:
2569:
2562:
2555:
2553:
2549:
2544:
2538:
2534:
2527:
2524:
2519:
2515:
2511:
2507:
2502:
2497:
2493:
2489:
2488:
2483:
2476:
2473:
2468:
2464:
2460:
2456:
2455:
2447:
2444:
2440:
2436:
2432:
2428:
2422:
2419:
2414:
2410:
2406:
2399:
2396:
2392:
2387:
2384:
2379:
2375:
2369:
2366:
2359:
2354:
2347:
2344:
2340:
2335:
2332:
2328:
2323:
2320:
2313:
2308:
2306:
2302:
2301:
2297:
2295:
2292:
2290:
2285:
2281:
2277:
2273:
2269:
2264:
2262:
2258:
2255:
2254:contravariant
2251:
2250:order complex
2247:
2243:
2239:
2235:
2228:
2223:
2221:
2217:
2216:domain theory
2213:
2209:
2206:and the name
2205:
2201:
2197:
2192:
2190:
2186:
2182:
2177:
2173:
2169:
2161:
2159:
2157:
2152:
2135:
2112:
2092:
2072:
2052:
2032:
2012:
1992:
1984:
1980:
1975:
1973:
1968:
1965:
1959:
1957:
1955:
1951:
1947:
1943:
1939:
1935:
1934:modal algebra
1931:
1927:
1924:
1919:
1918:
1915: ⊆
1914:
1906:
1903: ≤
1902:
1898:
1894:
1890:
1886:
1883:
1880:
1876:
1873: ≤
1872:
1868:
1864:
1860:
1856:
1853:
1852:
1851:
1849:
1848:
1843:
1842:
1837:
1833:
1829:
1828:
1819:
1817:
1811:
1810:
1805:
1804:
1799:
1798:
1793:
1792:
1788: :
1787:
1783:
1780:
1779:
1778:
1776:
1775:
1767:
1766:
1761:
1760:
1756: :
1755:
1752:
1751:
1750:
1745:
1744:
1739:
1738:
1733:
1732:
1727:
1726:
1722: :
1721:
1718:
1717:
1716:
1714:
1713:
1708:
1704:
1701: :
1700:
1699:
1694:
1693:
1688:
1684:
1681:is in fact a
1680:
1676:
1674:
1670:
1663:
1659:
1656: :
1655:
1654:
1650:
1647:
1643:
1640: :
1639:
1638:
1634:
1633:
1632:
1630:
1626:
1622:
1617:
1615:
1611:
1607:
1600:
1596:
1593: :
1592:
1591:
1587:
1584:
1580:
1577: :
1576:
1575:
1571:
1570:
1569:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1527:
1525:
1523:
1522:
1517:
1516:
1511:
1510:
1505:
1504:
1500: :
1499:
1496:
1492:
1491:
1484:
1480:
1478:
1474:
1473:
1468:
1460:
1459:
1454:
1453:
1448:
1447:
1442:
1441:
1436:
1432:
1431:
1427:
1426:
1425:
1420:
1419:
1414:
1413:
1408:
1405:
1404:
1403:
1400:
1398:
1394:
1390:
1389:
1384:
1376:
1375:
1370:
1369:
1364:
1363:
1358:
1357:
1352:
1348:
1347:
1343:
1342:
1341:
1339:
1338:
1333:
1329:
1325:
1321:
1317:
1316:
1311:
1308: ≤
1307:
1300:
1296:
1293: :
1292:
1289:
1288:
1287:
1282:
1281:
1276:
1275:
1271: :
1270:
1267:
1266:
1265:
1264:
1256:
1254:
1252:
1248:
1247:
1242:
1241:
1236:
1235:
1230:
1229:
1224:
1223:
1218:
1214:
1213:
1208:
1207:
1202:
1201:
1196:
1195:
1190:
1189:
1184:
1183:
1178:
1177:
1172:
1167:
1165:
1164:
1159:
1158:
1153:
1152:
1147:
1146:
1141:
1140:
1135:
1134:
1130:
1126:
1125:
1120:
1119:
1114:
1113:
1108:
1107:
1102:
1101:
1096:
1095:
1090:
1089:
1084:
1080:
1079:
1070:
1068:
1066:
1062:
1061:
1056:
1055:
1046:
1042:
1038:
1034:
1031:
1030:
1029:
1027:
1023:
1019:
1015:
1011:
1010:
1001:
979:
976:
973:
956:
953:
950:
944:
938:
935:
932:
920:
917:
914:
911:
908:
902:
899:
896:
893:
882:
881:
880:
879:
878:
876:
871:
854:
851:
848:
842:
799:
796:
793:
776:
773:
770:
764:
758:
755:
752:
740:
737:
734:
731:
728:
722:
719:
716:
713:
706:
703:
696:
695:
694:
692:
688:
672:
649:
646:
643:
637:
625:
617:
612:
610:
608:
605:is called an
604:
600:
596:
585:
581:
580:
575:
574:
568:
564:
563:
557:
553:
549:
545:
544:
540:
537:
534:
530:
529:
524:
523:
517:
513:
512:
506:
502:
498:
495:
492:
491:
486:
482:
479:
476:
473:
469:
465:
461:
457:
456:
451:
447:
443:
440:
439:
438:
435:
430:
426:
422:
418:
414:
410:
406:
405:
400:
396:
392:
389:
386:
382:
378:
374:
370:
369:
364:
361:
358:
354:
350:
346:
342:
341:upward closed
338:
337:
332:
329:
326:
325:
320:
316:
312:
308:
305:≤ satisfying
304:
300:
299:
294:
290:
286:
283:
280:
279:
278:
275:
270:
269:
264:
260:
257:
254:
253:
248:
244:
241:
240:
239:
236:
231:
230:
225:
221:
218:
215:
214:neighbourhood
211:
210:
205:
202:
201:
200:
197:
192:
191:
186:
183:
180:
179:
174:
171:
170:
169:
166:
165:
164:
162:
158:
154:
153:
144:
142:
140:
136:
132:
127:
125:
121:
117:
112:
110:
106:
105:determined by
101:
99:
95:
91:
87:
83:
79:
76:
72:
68:
63:
61:
57:
52:
50:
46:
42:
39:in which the
38:
34:
30:
19:
3308:Order theory
3080:
3074:& Orders
3052:Star product
2981:Well-founded
2934:Prefix order
2890:Distributive
2880:Complemented
2850:Foundational
2815:Completeness
2771:Zorn's lemma
2675:Cyclic order
2658:Key concepts
2628:Order theory
2586:
2580:
2571:
2567:
2533:Stone spaces
2532:
2526:
2491:
2485:
2475:
2458:
2452:
2446:
2438:
2434:
2426:
2421:
2412:
2408:
2398:
2386:
2377:
2368:
2346:
2334:
2329:, Theorem 7.
2322:
2304:
2293:
2283:
2275:
2272:modal frames
2265:
2241:
2224:
2207:
2193:
2171:
2165:
2153:
2085:included in
1976:
1969:
1966:
1963:
1929:
1920:
1916:
1912:
1910:
1904:
1900:
1896:
1892:
1888:
1884:
1878:
1874:
1870:
1866:
1862:
1858:
1854:
1846:
1845:
1840:
1839:
1826:
1825:
1823:
1815:
1808:
1807:
1802:
1801:
1796:
1795:
1790:
1789:
1785:
1781:
1773:
1772:
1770:
1764:
1763:
1758:
1757:
1753:
1748:
1742:
1741:
1736:
1735:
1730:
1729:
1724:
1723:
1719:
1711:
1710:
1706:
1702:
1697:
1696:
1691:
1690:
1686:
1678:
1677:
1672:
1667:are inverse
1666:
1661:
1657:
1652:
1651:
1645:
1641:
1636:
1635:
1628:
1620:
1618:
1609:
1603:
1598:
1594:
1589:
1588:
1582:
1578:
1573:
1572:
1557:
1545:
1533:
1531:
1520:
1519:
1514:
1513:
1508:
1507:
1502:
1501:
1497:
1495:identity map
1489:
1488:
1485:
1481:
1476:
1471:
1470:
1466:
1464:
1457:
1456:
1451:
1450:
1445:
1444:
1439:
1438:
1434:
1429:
1428:
1423:
1417:
1416:
1411:
1410:
1406:
1401:
1392:
1387:
1386:
1382:
1380:
1373:
1372:
1367:
1366:
1361:
1360:
1355:
1354:
1350:
1345:
1344:
1336:
1335:
1331:
1327:
1323:
1319:
1314:
1313:
1309:
1305:
1303:
1298:
1294:
1290:
1285:
1279:
1278:
1273:
1272:
1268:
1260:
1250:
1245:
1244:
1239:
1238:
1233:
1232:
1227:
1226:
1221:
1220:
1216:
1211:
1210:
1205:
1204:
1199:
1198:
1193:
1192:
1187:
1186:
1181:
1180:
1175:
1174:
1170:
1168:
1162:
1161:
1156:
1155:
1150:
1149:
1144:
1143:
1138:
1137:
1132:
1131:
1128:
1123:
1122:
1117:
1116:
1111:
1110:
1105:
1104:
1099:
1098:
1093:
1092:
1087:
1086:
1082:
1077:
1076:
1074:
1064:
1059:
1058:
1053:
1052:
1050:
1044:
1040:
1036:
1032:
1025:
1017:
1013:
1008:
1007:
1005:
872:
818:
686:
621:
606:
602:
598:
594:
592:
583:
578:
577:
572:
571:
566:
561:
560:
555:
551:
542:
541:
538:
527:
526:
521:
520:
515:
510:
509:
504:
500:
496:
489:
488:
480:
477:
471:
467:
463:
459:
454:
453:
449:
445:
441:
436:
428:
424:
420:
416:
412:
408:
403:
402:
398:
394:
390:
384:
380:
376:
372:
367:
366:
362:
356:
352:
348:
344:
335:
334:
331:Open up-set.
330:
323:
322:
318:
314:
310:
306:
297:
296:
284:
281:
276:
267:
266:
258:
251:
250:
242:
237:
228:
227:
219:
208:
207:
203:
198:
189:
188:
184:
177:
176:
172:
167:
160:
156:
151:
150:
148:
128:
113:
102:
97:
89:
81:
77:
64:
59:
55:
53:
48:
41:intersection
32:
26:
3258:Riesz space
3219:Isomorphism
3095:Normal cone
3017:Composition
2951:Semilattice
2860:Homogeneous
2845:Equivalence
2695:Total order
2574:(1): 17–25.
2431:Marcel Erné
2391:Arenas 1999
2339:Arenas 1999
2268:modal logic
2181:Øystein Ore
1938:modal frame
1623:denote the
1548:denote the
1536:denote the
576:where each
185:Closed set.
3297:Categories
3226:Order type
3160:Cofinality
3001:Well-order
2976:Transitive
2865:Idempotent
2798:Asymmetric
2415:: 501–518.
2327:Speer 2007
2314:References
1983:local base
1960:Properties
1556:; and let
1136:, we have
875:lower sets
691:upper sets
548:final hull
533:final sink
466:such that
415:such that
193:is closed.
86:upper sets
3277:Upper set
3214:Embedding
3150:Antichain
2971:Tolerance
2961:Symmetric
2956:Semiorder
2902:Reflexive
2820:Connected
2510:0002-9947
2363:Theorem 5
2358:0708.2136
1923:power set
1612:that are
1197:. Rather
1067:, ≤>.
1020:> the
977:∈
967:⟹
954:≤
945:∧
936:∈
918:∈
906:∀
897:⊆
858:⟩
855:τ
846:⟨
797:∈
787:⟹
774:≤
765:∧
756:∈
738:∈
726:∀
717:⊆
704:τ
673:τ
653:⟩
650:≤
641:⟨
173:Open set.
124:quotients
45:open sets
3072:Topology
2939:Preorder
2922:Eulerian
2885:Complete
2835:Directed
2825:Covering
2690:Preorder
2649:Category
2644:Glossary
2425:O. Ore,
2298:See also
1911:for all
1877:implies
1318:implies
1261:Given a
1063:) = <
622:Given a
559: :
508: :
485:coherent
444:A point
393:A point
343:i.e. if
303:preorder
181:is open.
71:preorder
37:topology
29:topology
3177:Duality
3155:Cofinal
3143:Related
3122:Fréchet
2999:)
2875:Bounded
2870:Lattice
2843:)
2841:Partial
2709:Results
2680:Lattice
2518:1994555
2435:Closure
2409:Mat. Sb
2257:lattice
2248:to the
2162:History
1948:from a
1936:from a
1784: ◦
1568:. Then
1409::
1395:) is a
1340:), let
531:form a
287:is the
92:are in
73:≤ on a
3202:Subnet
3182:Filter
3132:Normed
3117:Banach
3083:&
2990:Better
2927:Strict
2917:Graded
2808:topics
2639:Topics
2539:
2516:
2508:
2307:-space
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1830:, the
1771:where
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1487:space
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120:unions
49:finite
3192:Ideal
3170:Graph
2966:Total
2944:Total
2830:Dense
2564:(PDF)
2514:JSTOR
2353:arXiv
2025:, in
1671:over
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383:then
355:then
321:} in
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2783:list
2537:ISBN
2506:ISSN
2214:and
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1619:Let
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1564:and
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261:The
245:The
222:The
3197:Net
2997:Pre
2496:doi
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2463:doi
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483:is
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452:of
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58:or
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