102:
and statistical games, and defines and studies topological games within topology. After more than 35 years, the term “topological game” became widespread, and appeared in several hundreds of publications. The survey paper of Telgársky emphasizes the origin of topological games from the
70:, completeness and convergence properties, separation properties, covering and base properties, continuous images, Suslin sets, and singular spaces. At the same time, some topological properties that arise naturally in topological games can be generalized beyond a
733:
98:, the concept of “topological properties defined by games”, was introduced in the paper of Rastislav Telgársky, and later "spaces defined by topological games"; this approach is based on analogies with matrix games,
74:
context: by virtue of this duality, topological games have been widely used to describe new properties of topological spaces, and to put known properties under a different light. There are also close links with
1025:
962:. Play continues in this fashion, with players alternately picking a nonempty open subset of the previous play. After an infinite sequence of moves, one for each natural number, the game is finished, and
1263:
C. Berge, Topological games with perfect information. Contributions to the theory of games, vol. 3, 165–178. Annals of
Mathematics Studies, no. 39. Princeton University Press, Princeton, N. J., 1957.
321:
1294:
R. Telgársky, On topological properties defined by games, Topics in
Topology (Proc. Colloq. Keszthely 1972), Colloq. Math. Soc. János Bolyai, Vol. 8, North-Holland, Amsterdam 1974, 617–624.
960:
916:
386:
862:
The first topological game studied was the Banach–Mazur game, which is a motivating example of the connections between game-theoretic notions and topological properties.
800:
559:
412:
360:
157:
in his invited address . The number of moves in these games is always finite. The discovery or rediscovery of these topological games goes back to years 1948–49.
972:
62:
It turns out that some fundamental topological constructions have a natural counterpart in topological games; examples of these are the
51:
lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions like
1282:
283:
1348:
1353:
130:
1244:
1225:
1185:
925:
419:
857:
266:
104:
1069:
888:
1316:
1193:
123:
1205:
1169:
265:
The game is defined by the target property and the allowed moves at each step. For example, in the
76:
52:
28:
728:{\displaystyle s(\lambda ),J_{0},s(J_{0}),J_{1},s(J_{0},J_{1}),J_{2},s(J_{0},J_{1},J_{2}),\ldots }
365:
1209:
779:
48:
1180:
coreduction principle; separation and reduction properties of sets in close projective classes;
1154:
1096:
1058:
326:
This typical setup can be modified in various ways. For example, instead of being a subset of
99:
91:
32:
1229:
1217:
1146:
391:
134:
1272:
C. Berge, Théorie des jeux à n personnes, Mém. des Sc. Mat., Gauthier-Villars, Paris 1957.
333:
1329:
1201:
815:
167:
1121:
415:
119:
110:
There are two other meanings of topological games, but these are used less frequently.
71:
63:
44:
1235:
For a longer list and a more detailed account see the 1987 survey paper of Telgársky.
1342:
1181:
142:
1213:
1197:
1135:
87:
1328:
L. A. Petrosjan, Topological games and their applications to pursuit problems. I.
1303:
R. Telgársky, Spaces defined by topological games, Fund. Math. 88 (1975), 193–223.
56:
1176:
Many more games have been introduced over the years, to study, among others: the
1030:
The game-theoretic and topological connections demonstrated by the game include:
1221:
1128:
1117:
1050:
276:), the allowed moves are nonempty open subsets of the previous move, and player
138:
67:
40:
20:
1189:
1177:
1081:
1054:
154:
36:
126:
games. The trajectories in these topological games are continuous in time.
146:
1317:"Topological Games: On the 50th Anniversary of the Banach-Mazur Game"
35:. Players choose objects with topological properties such as points,
491:
is a function defined over every legal finite sequence of moves of
1020:{\displaystyle X\cap \bigcap _{n\in \omega }I_{n}\neq \emptyset .}
414:. Alternatively, the sequence of moves might have length some
90:, who defined the basic ideas and formalism in analogy with
1168:
chooses a member or finite subset of that collection. See
1208:. Topological games have also been related to ideas in
542:
on the sequence of their opponent's prior moves. So if
181:, who alternately pick subsets of a topological space
975:
928:
891:
782:
562:
394:
368:
336:
286:
47:. Time is generally discrete, but the plays may have
843:
if it depends both on the last move of the opponent
1037:has a winning strategy in the game if and only if
1019:
954:
910:
885:begins the game by picking a nonempty open subset
794:
727:
406:
380:
354:
315:
145:games (projective plane games), and Gale's games (
173:The typical setup is a game between two players,
1311:
1309:
818:that there are non-determined topological games.
316:{\displaystyle \bigcap _{n}I_{n}\neq \emptyset }
1220:, infinite strings of alternating quantifiers,
743:. (Here λ denotes the empty sequence of moves.)
495:s opponent. For example, a strategy for player
802:. If either player has a winning strategy for
258:satisfies some property, and otherwise player
1319:, Rocky Mountain J. Math. 17 (1987), 227–276.
8:
1170:Selection principle § Topological games
215:. There is a round for every natural number
166:Many frameworks can be defined for infinite
1124:— a modification of the Banach–Mazur game;
1112:Some other notable topological games are:
478:is either a win or a loss for each player.
219:, and after all rounds are played, player
1002:
986:
974:
946:
933:
927:
896:
890:
781:
710:
697:
684:
665:
649:
636:
617:
601:
582:
561:
393:
367:
335:
301:
291:
285:
756:if for every play according to strategy
435:of the game is a sequence of legal moves
206:, and player II responds with a subset
16:Mathematical game on a topological space
1256:
1164:chooses a (topological) collection and
829:if it depends only on the last move by
1141:the point-open game — in which player
1131:— played on a subset of the real line;
1076:has a winning strategy if and only if
922:responds with a nonempty open subset
764:, for any sequence of legal moves by
7:
1160:selection games — each round player
955:{\displaystyle J_{0}\subseteq I_{0}}
330:, each move might consist of a pair
1281:A. R. Pears, On topological games,
1011:
847:on the ordinal number of the move.
310:
162:Basic setup for a topological game
14:
1084:in some nonempty open subset of
1057:if it is the countable union of
911:{\displaystyle I_{0}\subseteq Y}
869:be a topological space, and let
772:has a winning strategy for game
31:played between two players on a
1103:, then the game is determined.
786:
716:
677:
655:
629:
607:
594:
572:
566:
530:. A game is said to be played
349:
337:
1:
1138:— related to siftable spaces;
122:in the study of antagonistic
1283:Proc. Cambridge Philos. Soc.
760:results in a win for player
381:{\displaystyle I\subseteq X}
795:{\displaystyle P\uparrow G}
1370:
855:
833:s opponent; a strategy is
94:. A different meaning for
546:is a strategy for player
1218:infinitely-long formulas
426:Definitions and notation
170:of perfect information.
86:was first introduced by
1108:Other topological games
741:according to strategy s
532:according to strategy s
27:is an infinite game of
1226:partially ordered sets
1186:descriptive set theory
1021:
956:
912:
796:
768:s opponent. If player
748:A strategy for player
729:
408:
407:{\displaystyle p\in x}
382:
356:
317:
1070:complete metric space
1022:
957:
913:
852:The Banach–Mazur game
797:
730:
538:move is the value of
409:
383:
357:
355:{\displaystyle (I,p)}
318:
223:wins if the sequence
141:games (Y games), the
1232:of infinite graphs.
1206:computable functions
1194:closed graph theorem
973:
966:wins if and only if
926:
889:
814:It follows from the
780:
560:
392:
366:
334:
284:
77:selection principles
1332:10 (1972), 194–202.
1285:61 (1965), 165–171.
149:games) were called
53:topological closure
29:perfect information
1245:Topological puzzle
1210:mathematical logic
1184:sieves; invariant
1155:open neighborhoods
1059:nowhere-dense sets
1017:
997:
952:
908:
792:
776:, this is denoted
725:
404:
378:
352:
313:
296:
100:differential games
92:topological groups
1349:Topological games
1200:; MP-spaces; the
1097:property of Baire
1049:(a set is of the
982:
858:Banach–Mazur game
287:
267:Banach–Mazur game
189:th round, player
151:topological games
105:Banach–Mazur game
33:topological space
1361:
1354:General topology
1333:
1326:
1320:
1313:
1304:
1301:
1295:
1292:
1286:
1279:
1273:
1270:
1264:
1261:
1230:chromatic number
1026:
1024:
1023:
1018:
1007:
1006:
996:
961:
959:
958:
953:
951:
950:
938:
937:
917:
915:
914:
909:
901:
900:
840:
839:
801:
799:
798:
793:
734:
732:
731:
726:
715:
714:
702:
701:
689:
688:
670:
669:
654:
653:
641:
640:
622:
621:
606:
605:
587:
586:
534:if every player
526:) to subsets of
503:from sequences (
476:result of a play
413:
411:
410:
405:
387:
385:
384:
379:
361:
359:
358:
353:
322:
320:
319:
314:
306:
305:
295:
168:positional games
116:topological game
96:topological game
84:topological game
25:topological game
1369:
1368:
1364:
1363:
1362:
1360:
1359:
1358:
1339:
1338:
1337:
1336:
1330:SIAM J. Control
1327:
1323:
1314:
1307:
1302:
1298:
1293:
1289:
1280:
1276:
1271:
1267:
1262:
1258:
1253:
1241:
1202:axiom of choice
1110:
998:
971:
970:
942:
929:
924:
923:
892:
887:
886:
873:be a subset of
860:
854:
837:
836:
821:A strategy for
816:axiom of choice
778:
777:
706:
693:
680:
661:
645:
632:
613:
597:
578:
558:
557:
525:
516:
509:
468:
461:
454:
447:
428:
390:
389:
364:
363:
332:
331:
297:
282:
281:
253:
246:
239:
232:
214:
201:
193:plays a subset
164:
124:pursuit–evasion
17:
12:
11:
5:
1367:
1365:
1357:
1356:
1351:
1341:
1340:
1335:
1334:
1321:
1315:R. Telgársky,
1305:
1296:
1287:
1274:
1265:
1255:
1254:
1252:
1249:
1248:
1247:
1240:
1237:
1174:
1173:
1158:
1139:
1132:
1125:
1120:introduced by
1109:
1106:
1105:
1104:
1089:
1062:
1051:first category
1043:first category
1028:
1027:
1016:
1013:
1010:
1005:
1001:
995:
992:
989:
985:
981:
978:
949:
945:
941:
936:
932:
907:
904:
899:
895:
856:Main article:
853:
850:
849:
848:
819:
810:is said to be
791:
788:
785:
752:is said to be
745:
744:
737:
736:
735:
724:
721:
718:
713:
709:
705:
700:
696:
692:
687:
683:
679:
676:
673:
668:
664:
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648:
644:
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635:
631:
628:
625:
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616:
612:
609:
604:
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596:
593:
590:
585:
581:
577:
574:
571:
568:
565:
552:
551:
521:
514:
507:
499:is a function
480:
479:
472:
471:
470:
466:
459:
452:
445:
437:
436:
427:
424:
416:ordinal number
403:
400:
397:
377:
374:
371:
351:
348:
345:
342:
339:
312:
309:
304:
300:
294:
290:
256:
255:
251:
244:
237:
230:
210:
197:
163:
160:
159:
158:
127:
120:Leon Petrosjan
118:introduced by
72:game-theoretic
64:Baire property
45:open coverings
15:
13:
10:
9:
6:
4:
3:
2:
1366:
1355:
1352:
1350:
1347:
1346:
1344:
1331:
1325:
1322:
1318:
1312:
1310:
1306:
1300:
1297:
1291:
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1278:
1275:
1269:
1266:
1260:
1257:
1250:
1246:
1243:
1242:
1238:
1236:
1233:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1198:webbed spaces
1195:
1191:
1187:
1183:
1179:
1171:
1167:
1163:
1159:
1156:
1152:
1148:
1144:
1140:
1137:
1133:
1130:
1126:
1123:
1119:
1115:
1114:
1113:
1107:
1102:
1098:
1094:
1090:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1033:
1032:
1031:
1014:
1008:
1003:
999:
993:
990:
987:
983:
979:
976:
969:
968:
967:
965:
947:
943:
939:
934:
930:
921:
918:, and player
905:
902:
897:
893:
884:
880:
877:, called the
876:
872:
868:
863:
859:
851:
846:
842:
841:
832:
828:
824:
820:
817:
813:
809:
805:
789:
783:
775:
771:
767:
763:
759:
755:
751:
747:
746:
742:
738:
722:
719:
711:
707:
703:
698:
694:
690:
685:
681:
674:
671:
666:
662:
658:
650:
646:
642:
637:
633:
626:
623:
618:
614:
610:
602:
598:
591:
588:
583:
579:
575:
569:
563:
556:
555:
554:
553:
549:
545:
541:
537:
533:
529:
524:
520:
513:
506:
502:
498:
494:
490:
486:
482:
481:
477:
473:
465:
458:
451:
444:
441:
440:
439:
438:
434:
430:
429:
425:
423:
421:
417:
401:
398:
395:
375:
372:
369:
346:
343:
340:
329:
324:
307:
302:
298:
292:
288:
279:
275:
271:
268:
263:
261:
250:
243:
236:
229:
226:
225:
224:
222:
218:
213:
209:
205:
200:
196:
192:
188:
184:
180:
176:
171:
169:
161:
156:
152:
148:
144:
140:
136:
132:
129:The games of
128:
125:
121:
117:
113:
112:
111:
108:
106:
101:
97:
93:
89:
85:
80:
78:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1324:
1299:
1290:
1277:
1268:
1259:
1234:
1222:ultrafilters
1214:model theory
1175:
1165:
1161:
1150:
1142:
1136:Choquet game
1111:
1100:
1092:
1085:
1077:
1073:
1065:
1046:
1042:
1038:
1034:
1029:
963:
919:
882:
878:
874:
870:
866:
864:
861:
844:
835:
834:
830:
826:
822:
811:
807:
803:
773:
769:
765:
761:
757:
753:
749:
740:
547:
543:
539:
535:
531:
527:
522:
518:
511:
504:
500:
496:
492:
488:
484:
475:
463:
456:
449:
442:
432:
327:
325:
277:
273:
269:
264:
259:
257:
248:
241:
234:
227:
220:
216:
211:
207:
203:
198:
194:
190:
186:
182:
178:
174:
172:
165:
150:
115:
109:
95:
88:Claude Berge
83:
81:
68:Baire spaces
61:
24:
18:
1190:Suslin sets
1149:and player
1129:Banach game
1118:binary game
879:winning set
812:determined.
487:for player
418:other than
57:convergence
49:transfinite
41:closed sets
21:mathematics
1343:Categories
1251:References
1228:, and the
1178:Kuratowski
1041:is of the
827:stationary
550:, the play
155:David Gale
1012:∅
1009:≠
994:ω
991:∈
984:⋂
980:∩
940:⊆
903:⊆
881:. Player
787:↑
723:…
570:λ
399:∈
373:⊆
311:∅
308:≠
289:⋂
185:. In the
135:Hex games
114:The term
82:The term
37:open sets
1239:See also
1157:of them;
1153:chooses
1145:chooses
1095:has the
1082:comeagre
485:strategy
280:wins if
147:Bridg-It
1072:, then
806:, then
754:winning
517:, ...,
143:Shapley
137:), the
1192:; the
1147:points
1055:meagre
838:Markov
362:where
262:wins.
139:Milnor
1182:Luzin
1068:is a
133:(the
1134:the
1127:the
1122:Ulam
1116:the
865:Let
474:The
469:,...
433:play
388:and
254:,...
177:and
131:Nash
55:and
43:and
23:, a
1099:in
1091:If
1080:is
1064:If
1053:or
1045:in
845:and
825:is
739:is
202:of
153:by
19:In
1345::
1308:^
1224:,
1216:,
1212:,
1204:;
1196:;
1188:;
1166:II
1151:II
1061:).
1035:II
920:II
831:P'
766:P'
510:,
493:P'
483:A
462:,
455:,
448:,
431:A
422:.
323:.
270:BM
260:II
247:,
240:,
233:,
179:II
107:.
79:.
66:,
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1172:.
1162:I
1143:I
1101:Y
1093:X
1088:.
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1078:X
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1066:Y
1047:Y
1039:X
1015:.
1004:n
1000:I
988:n
977:X
964:I
948:0
944:I
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931:J
906:Y
898:0
894:I
883:I
875:Y
871:X
867:Y
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808:G
804:G
790:G
784:P
774:G
770:P
762:P
758:s
750:P
720:,
717:)
712:2
708:J
704:,
699:1
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691:,
686:0
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678:(
675:s
672:,
667:2
663:J
659:,
656:)
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647:J
643:,
638:0
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630:(
627:s
624:,
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611:,
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576:,
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567:(
564:s
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544:s
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512:J
508:0
505:J
501:s
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344:,
341:I
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238:0
235:J
231:0
228:I
221:I
217:n
212:n
208:J
204:X
199:n
195:I
191:I
187:n
183:X
175:I
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