1996:
1779:
2017:
1985:
2054:
2027:
2007:
313:
his property is strictly stronger than the Menger property. In 2002, Chaber and Pol in unpublished note, using dichotomy proof, showed that there is a
Hurewicz subset of the real line that is not Menger. In 2008, Tsaban and Zdomskyy gave a uniform example of a Hurewicz subset of the real line that is
290:, and Szeptycki proved that Hurewicz's conjecture is false, by showing that there is, in ZFC, a set of real numbers that is Menger but not σ-compact. Their proof was dichotomic, and the set witnessing the failure of the conjecture heavily depends on whether a certain (undecidable) axiom holds or not.
172:
679:
is unbounded if it is not bounded. Hurewicz proved that a subset of the real line is
Hurewicz iff every continuous image of that space into the Baire space is unbounded. In particular, every subset of the real line of cardinality less than the
1617:
1051:
232:
84:
963:
858:
402:
587:
677:
550:
359:
89:
1293:
895:
790:
1513:
Boaz Tsaban (2011), 'Menger's and
Hurewicz's Problems: Solutions from "The Book" and refinements', in "Set Theory and its Applications" Contemporary Mathematics 533, 211–226.
704:
479:
1345:
1181:
1087:
620:
435:
1241:
272:
646:
254:
is true, and whether his property is strictly stronger than the Menger property, but he conjectured that in the class of metric spaces his property is equivalent to
1261:
1221:
1201:
1154:
1134:
1114:
991:
915:
810:
753:
733:
519:
499:
2057:
996:
177:
1528:
2078:
35:
920:
815:
1056:
A player has a winning strategy if he knows how to play in order to win the game (formally, a winning strategy is a function).
1420:
Just, Winfried; Miller, Arnold W.; Scheepers, Marion; Szeptycki, Paul J. (1996-11-11). "The combinatorics of open covers II".
301:'s solution based on their work ) gave a uniform ZFC example of a Hurewicz subset of the real line that is not σ-compact.
364:
1691:
555:
2045:
2040:
2035:
651:
524:
333:
167:{\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subset {\mathcal {U}}_{2},\ldots }
1937:
1587:
1469:
1422:
1059:
A topological space is
Hurewicz iff Alice has no winning strategy in the Hurewicz game played on this space.
327:
For subsets of the real line, the
Hurewicz property can be characterized using continuous functions into the
1266:
869:
764:
1945:
1374:
685:
1744:
440:
2030:
2016:
1323:
1159:
1065:
592:
407:
1965:
1886:
1763:
1751:
1724:
1684:
1062:
1960:
681:
1807:
1734:
1226:
257:
1526:
Tsaban, Boaz; Zdomskyy, Lyubomyr (2008-01-01). "Scales, fields, and a problem of
Hurewicz".
625:
1955:
1907:
1881:
1729:
8:
1802:
1467:
Bartoszynski, Tomek; Shelah, Saharon (2001-11-15). "Continuous images of sets of reals".
25:
2006:
2000:
1970:
1950:
1871:
1861:
1739:
1719:
1655:
1629:
1563:
1537:
1496:
1478:
1449:
1431:
1399:
1246:
1206:
1186:
1139:
1119:
1099:
976:
900:
795:
738:
718:
504:
484:
246:
introduced the above property of topological spaces that is formally stronger than the
1582:
1492:
1445:
29:
2083:
1995:
1988:
1854:
1812:
1677:
1647:
1601:
1555:
1403:
1391:
21:
2020:
1659:
1567:
1500:
1453:
1312:
1768:
1714:
1639:
1596:
1547:
1488:
1441:
1383:
287:
1827:
1822:
1351:
1094:
243:
2010:
1372:
1917:
1849:
294:
2072:
1927:
1837:
1817:
1651:
1559:
1395:
1046:{\displaystyle \bigcup {\mathcal {F}}_{1},\bigcup {\mathcal {F}}_{2},\ldots }
227:{\displaystyle \bigcup {\mathcal {F}}_{1},\bigcup {\mathcal {F}}_{2},\ldots }
1912:
1832:
1778:
1308:
251:
247:
1922:
328:
298:
174:
such that every point of the space belongs to all but finitely many sets
32:. A Hurewicz space is a space in which for every sequence of open covers
1643:
1866:
1797:
1756:
1551:
1387:
1891:
1542:
1483:
1436:
1876:
1844:
1793:
1700:
1616:
Chodounský, David; Repovš, Dušan; Zdomskyy, Lyubomyr (2015-12-01).
1634:
1618:"Mathias Forcing and Combinatorial Covering Properties of Filters"
1514:
79:{\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots }
1419:
958:{\displaystyle {\mathcal {F}}_{2}\subset {\mathcal {U}}_{2}}
853:{\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1}}
1669:
1053:, then Bob wins the Hurewicz game. Otherwise, Alice wins.
310:
283:
1615:
286:
every
Hurewicz metric space is σ-compact. Just, Miller,
1326:
1304:
Every compact, and even σ-compact, space is
Hurewicz.
1269:
1249:
1229:
1209:
1189:
1162:
1142:
1122:
1102:
1068:
999:
979:
923:
903:
872:
818:
798:
767:
741:
721:
688:
654:
628:
595:
558:
527:
507:
487:
443:
410:
367:
336:
260:
180:
92:
38:
735:
be a topological space. The
Hurewicz game played on
710:
1466:
1339:
1287:
1255:
1235:
1215:
1195:
1175:
1148:
1128:
1108:
1081:
1045:
985:
957:
909:
889:
852:
804:
784:
747:
727:
698:
671:
640:
614:
581:
544:
513:
493:
473:
429:
397:{\displaystyle f,g\in \mathbb {N} ^{\mathbb {N} }}
396:
353:
266:
226:
166:
78:
322:
2070:
1350:Hurewicz's property characterizes filters whose
1317:Continuous image of a Hurewicz space is Hurewicz
582:{\displaystyle g\in \mathbb {N} ^{\mathbb {N} }}
1583:"Combinatorics of open covers I: Ramsey theory"
1525:
1685:
1320:The Hurewicz property is closed under taking
1529:Journal of the European Mathematical Society
672:{\displaystyle \mathbb {N} ^{\mathbb {N} }}
545:{\displaystyle \mathbb {N} ^{\mathbb {N} }}
354:{\displaystyle \mathbb {N} ^{\mathbb {N} }}
2053:
2026:
1692:
1678:
755:is a game with two players Alice and Bob.
481:for all but finitely many natural numbers
1633:
1600:
1580:
1541:
1482:
1435:
663:
657:
573:
567:
536:
530:
388:
382:
345:
339:
1371:
1354:notion does not add unbounded functions.
1116:is Hurewicz iff for every compact space
277:
2071:
993:belongs to all but finitely many sets
1673:
304:
1574:
1415:
1413:
317:
1288:{\displaystyle X\subset Y\subset G}
691:
86:of the space there are finite sets
13:
1026:
1006:
944:
927:
890:{\displaystyle {\mathcal {U}}_{2}}
876:
839:
822:
785:{\displaystyle {\mathcal {U}}_{1}}
771:
552:is bounded if there is a function
207:
187:
147:
130:
113:
96:
59:
42:
14:
2095:
1410:
711:Topological game characterization
2079:Properties of topological spaces
2052:
2025:
2015:
2005:
1994:
1984:
1983:
1777:
1515:https://arxiv.org/abs/0909.5645
699:{\displaystyle {\mathfrak {b}}}
24:that satisfies a certain basic
1609:
1519:
1507:
1460:
1365:
1089:-neighborhood characterization
866:: Alice chooses an open cover
761:: Alice chooses an open cover
468:
462:
453:
447:
323:Combinatorial characterization
1:
1622:The Journal of Symbolic Logic
1588:Topology and Its Applications
1493:10.1016/S0166-8641(00)00079-1
1470:Topology and Its Applications
1446:10.1016/S0166-8641(96)00075-2
1423:Topology and Its Applications
1358:
1298:
474:{\displaystyle f(n)\leq g(n)}
282:Hurewicz conjectured that in
1699:
1602:10.1016/0166-8641(95)00067-4
973:If every point of the space
7:
1340:{\displaystyle F_{\sigma }}
1176:{\displaystyle G_{\delta }}
1082:{\displaystyle G_{\delta }}
917:. Bob chooses a finite set
812:. Bob chooses a finite set
615:{\displaystyle f\leq ^{*}g}
430:{\displaystyle f\leq ^{*}g}
10:
2100:
1946:Banach fixed-point theorem
1581:Scheepers, Marion (1996).
1307:Every Hurewicz space is a
309:Hurewicz asked whether in
237:
1979:
1936:
1900:
1786:
1775:
1707:
1375:Mathematische Zeitschrift
314:Menger but not Hurewicz.
250:. He didn't know whether
1236:{\displaystyle \sigma }
267:{\displaystyle \sigma }
2001:Mathematics portal
1901:Metrics and properties
1887:Second-countable space
1341:
1289:
1257:
1237:
1217:
1197:
1177:
1150:
1130:
1110:
1083:
1047:
987:
959:
911:
891:
854:
806:
786:
749:
729:
700:
673:
642:
641:{\displaystyle f\in A}
616:
583:
546:
515:
495:
475:
431:
398:
355:
268:
228:
168:
80:
1342:
1290:
1258:
1238:
1218:
1203:containing the space
1198:
1178:
1151:
1136:containing the space
1131:
1111:
1084:
1048:
988:
960:
912:
892:
855:
807:
787:
750:
730:
701:
674:
643:
617:
584:
547:
516:
496:
476:
432:
399:
356:
278:Hurewicz's conjecture
269:
229:
169:
81:
1956:Invariance of domain
1908:Euler characteristic
1882:Bundle (mathematics)
1324:
1267:
1247:
1227:
1207:
1187:
1160:
1140:
1120:
1100:
1066:
997:
977:
921:
901:
870:
816:
796:
765:
739:
719:
686:
652:
626:
593:
556:
525:
505:
485:
441:
408:
365:
334:
258:
178:
90:
36:
1966:Tychonoff's theorem
1961:Poincaré conjecture
1715:General (point-set)
1644:10.1017/jsl.2014.73
1311:, and thus it is a
252:Menger's conjecture
26:selection principle
1951:De Rham cohomology
1872:Polyhedral complex
1862:Simplicial complex
1388:10.1007/BF01216792
1337:
1285:
1253:
1233:
1213:
1193:
1173:
1146:
1126:
1106:
1079:
1043:
983:
955:
907:
887:
850:
802:
782:
745:
725:
696:
669:
638:
622:for all functions
612:
579:
542:
511:
491:
471:
427:
394:
351:
305:Hurewicz's problem
264:
224:
164:
76:
16:In mathematics, a
2066:
2065:
1855:fundamental group
1256:{\displaystyle Y}
1216:{\displaystyle X}
1196:{\displaystyle C}
1149:{\displaystyle X}
1129:{\displaystyle C}
1109:{\displaystyle X}
986:{\displaystyle X}
910:{\displaystyle X}
805:{\displaystyle X}
748:{\displaystyle X}
728:{\displaystyle X}
514:{\displaystyle A}
494:{\displaystyle n}
318:Characterizations
293:Bartoszyński and
28:that generalizes
22:topological space
2091:
2056:
2055:
2029:
2028:
2019:
2009:
1999:
1998:
1987:
1986:
1781:
1694:
1687:
1680:
1671:
1670:
1664:
1663:
1637:
1628:(4): 1398–1410.
1613:
1607:
1606:
1604:
1578:
1572:
1571:
1552:10.4171/jems/132
1545:
1523:
1517:
1511:
1505:
1504:
1486:
1464:
1458:
1457:
1439:
1417:
1408:
1407:
1369:
1346:
1344:
1343:
1338:
1336:
1335:
1294:
1292:
1291:
1286:
1262:
1260:
1259:
1254:
1242:
1240:
1239:
1234:
1222:
1220:
1219:
1214:
1202:
1200:
1199:
1194:
1182:
1180:
1179:
1174:
1172:
1171:
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1115:
1113:
1112:
1107:
1088:
1086:
1085:
1080:
1078:
1077:
1052:
1050:
1049:
1044:
1036:
1035:
1030:
1029:
1016:
1015:
1010:
1009:
992:
990:
989:
984:
964:
962:
961:
956:
954:
953:
948:
947:
937:
936:
931:
930:
916:
914:
913:
908:
896:
894:
893:
888:
886:
885:
880:
879:
859:
857:
856:
851:
849:
848:
843:
842:
832:
831:
826:
825:
811:
809:
808:
803:
791:
789:
788:
783:
781:
780:
775:
774:
754:
752:
751:
746:
734:
732:
731:
726:
705:
703:
702:
697:
695:
694:
682:bounding number
678:
676:
675:
670:
668:
667:
666:
660:
647:
645:
644:
639:
621:
619:
618:
613:
608:
607:
588:
586:
585:
580:
578:
577:
576:
570:
551:
549:
548:
543:
541:
540:
539:
533:
520:
518:
517:
512:
500:
498:
497:
492:
480:
478:
477:
472:
436:
434:
433:
428:
423:
422:
403:
401:
400:
395:
393:
392:
391:
385:
361:. For functions
360:
358:
357:
352:
350:
349:
348:
342:
273:
271:
270:
265:
233:
231:
230:
225:
217:
216:
211:
210:
197:
196:
191:
190:
173:
171:
170:
165:
157:
156:
151:
150:
140:
139:
134:
133:
123:
122:
117:
116:
106:
105:
100:
99:
85:
83:
82:
77:
69:
68:
63:
62:
52:
51:
46:
45:
2099:
2098:
2094:
2093:
2092:
2090:
2089:
2088:
2069:
2068:
2067:
2062:
1993:
1975:
1971:Urysohn's lemma
1932:
1896:
1782:
1773:
1745:low-dimensional
1703:
1698:
1668:
1667:
1614:
1610:
1579:
1575:
1524:
1520:
1512:
1508:
1465:
1461:
1418:
1411:
1370:
1366:
1361:
1352:Mathias forcing
1331:
1327:
1325:
1322:
1321:
1301:
1268:
1265:
1264:
1248:
1245:
1244:
1228:
1225:
1224:
1208:
1205:
1204:
1188:
1185:
1184:
1167:
1163:
1161:
1158:
1157:
1141:
1138:
1137:
1121:
1118:
1117:
1101:
1098:
1097:
1095:Tychonoff space
1091:
1073:
1069:
1067:
1064:
1063:
1031:
1025:
1024:
1023:
1011:
1005:
1004:
1003:
998:
995:
994:
978:
975:
974:
949:
943:
942:
941:
932:
926:
925:
924:
922:
919:
918:
902:
899:
898:
881:
875:
874:
873:
871:
868:
867:
844:
838:
837:
836:
827:
821:
820:
819:
817:
814:
813:
797:
794:
793:
776:
770:
769:
768:
766:
763:
762:
740:
737:
736:
720:
717:
716:
713:
690:
689:
687:
684:
683:
662:
661:
656:
655:
653:
650:
649:
627:
624:
623:
603:
599:
594:
591:
590:
572:
571:
566:
565:
557:
554:
553:
535:
534:
529:
528:
526:
523:
522:
506:
503:
502:
486:
483:
482:
442:
439:
438:
418:
414:
409:
406:
405:
387:
386:
381:
380:
366:
363:
362:
344:
343:
338:
337:
335:
332:
331:
325:
320:
307:
280:
259:
256:
255:
248:Menger property
244:Witold Hurewicz
240:
212:
206:
205:
204:
192:
186:
185:
184:
179:
176:
175:
152:
146:
145:
144:
135:
129:
128:
127:
118:
112:
111:
110:
101:
95:
94:
93:
91:
88:
87:
64:
58:
57:
56:
47:
41:
40:
39:
37:
34:
33:
12:
11:
5:
2097:
2087:
2086:
2081:
2064:
2063:
2061:
2060:
2050:
2049:
2048:
2043:
2038:
2023:
2013:
2003:
1991:
1980:
1977:
1976:
1974:
1973:
1968:
1963:
1958:
1953:
1948:
1942:
1940:
1934:
1933:
1931:
1930:
1925:
1920:
1918:Winding number
1915:
1910:
1904:
1902:
1898:
1897:
1895:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1858:
1857:
1852:
1850:homotopy group
1842:
1841:
1840:
1835:
1830:
1825:
1820:
1810:
1805:
1800:
1790:
1788:
1784:
1783:
1776:
1774:
1772:
1771:
1766:
1761:
1760:
1759:
1749:
1748:
1747:
1737:
1732:
1727:
1722:
1717:
1711:
1709:
1705:
1704:
1697:
1696:
1689:
1682:
1674:
1666:
1665:
1608:
1573:
1536:(3): 837–866.
1518:
1506:
1477:(2): 243–253.
1459:
1430:(3): 241–266.
1409:
1382:(1): 401–421.
1363:
1362:
1360:
1357:
1356:
1355:
1348:
1334:
1330:
1318:
1315:
1313:Lindelöf space
1305:
1300:
1297:
1284:
1281:
1278:
1275:
1272:
1252:
1232:
1212:
1192:
1170:
1166:
1145:
1125:
1105:
1090:
1076:
1072:
1061:
1042:
1039:
1034:
1028:
1022:
1019:
1014:
1008:
1002:
982:
952:
946:
940:
935:
929:
906:
884:
878:
847:
841:
835:
830:
824:
801:
779:
773:
744:
724:
712:
709:
693:
665:
659:
648:. A subset of
637:
634:
631:
611:
606:
602:
598:
575:
569:
564:
561:
538:
532:
510:
490:
470:
467:
464:
461:
458:
455:
452:
449:
446:
426:
421:
417:
413:
390:
384:
379:
376:
373:
370:
347:
341:
324:
321:
319:
316:
306:
303:
279:
276:
274:-compactness.
263:
239:
236:
223:
220:
215:
209:
203:
200:
195:
189:
183:
163:
160:
155:
149:
143:
138:
132:
126:
121:
115:
109:
104:
98:
75:
72:
67:
61:
55:
50:
44:
18:Hurewicz space
9:
6:
4:
3:
2:
2096:
2085:
2082:
2080:
2077:
2076:
2074:
2059:
2051:
2047:
2044:
2042:
2039:
2037:
2034:
2033:
2032:
2024:
2022:
2018:
2014:
2012:
2008:
2004:
2002:
1997:
1992:
1990:
1982:
1981:
1978:
1972:
1969:
1967:
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1928:Orientability
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1725:Combinatorial
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1378:(in German).
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1243:-compact set
1230:
1223:, there is a
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1103:
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828:
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742:
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708:
707:is Hurewicz.
706:
635:
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629:
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31:
30:σ-compactness
27:
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19:
2058:Publications
1923:Chern number
1913:Betti number
1796: /
1787:Key concepts
1735:Differential
1625:
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1543:math/0507043
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1484:math/0001051
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1462:
1437:math/9509211
1427:
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1309:Menger space
1183:subset G of
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17:
15:
2021:Wikiversity
1938:Key results
501:. A subset
329:Baire space
2073:Categories
1867:CW complex
1808:Continuity
1798:Closed set
1757:cohomology
1359:References
1299:Properties
589:such that
297:(see also
2046:geometric
2041:algebraic
1892:Cobordism
1828:Hausdorff
1823:connected
1740:Geometric
1730:Continuum
1720:Algebraic
1652:0022-4812
1635:1401.2283
1595:: 31–62.
1560:1435-9855
1404:119867793
1396:0025-5874
1333:σ
1280:⊂
1274:⊂
1231:σ
1169:δ
1075:δ
1041:…
1021:⋃
1001:⋃
939:⊂
864:2nd round
834:⊂
759:1st round
633:∈
605:∗
601:≤
563:∈
457:≤
420:∗
416:≤
378:∈
288:Scheepers
262:σ
242:In 1926,
222:…
202:⋃
182:⋃
162:…
142:⊂
108:⊂
74:…
2084:Topology
2011:Wikibook
1989:Category
1877:Manifold
1845:Homotopy
1803:Interior
1794:Open set
1752:Homology
1701:Topology
1660:15867466
1568:13902742
1501:14343145
1454:14946860
1156:, and a
404:, write
2036:general
1838:uniform
1818:compact
1769:Digital
1347:subsets
238:History
2031:Topics
1833:metric
1708:Fields
1658:
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1566:
1558:
1499:
1452:
1402:
1394:
299:Tsaban
295:Shelah
1813:Space
1656:S2CID
1630:arXiv
1564:S2CID
1538:arXiv
1497:S2CID
1479:arXiv
1450:S2CID
1432:arXiv
1400:S2CID
1263:with
20:is a
1648:ISSN
1556:ISSN
1392:ISSN
969:etc.
715:Let
1640:doi
1597:doi
1548:doi
1489:doi
1475:116
1442:doi
1384:doi
897:of
792:of
521:of
437:if
311:ZFC
284:ZFC
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1626:80
1624:.
1620:.
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1532:.
1495:.
1487:.
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1448:.
1440:.
1428:73
1426:.
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1398:.
1390:.
1380:24
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1093:A
965:.
860:.
234:.
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1550::
1540::
1503:.
1491::
1481::
1456:.
1444::
1434::
1406:.
1386::
1329:F
1283:G
1277:Y
1271:X
1251:Y
1211:X
1191:C
1165:G
1144:X
1124:C
1104:X
1071:G
1038:,
1033:2
1027:F
1018:,
1013:1
1007:F
981:X
951:2
945:U
934:2
928:F
905:X
883:2
877:U
846:1
840:U
829:1
823:F
800:X
778:1
772:U
743:X
723:X
692:b
664:N
658:N
636:A
630:f
610:g
597:f
574:N
568:N
560:g
537:N
531:N
509:A
489:n
469:)
466:n
463:(
460:g
454:)
451:n
448:(
445:f
425:g
412:f
389:N
383:N
375:g
372:,
369:f
346:N
340:N
219:,
214:2
208:F
199:,
194:1
188:F
159:,
154:2
148:U
137:2
131:F
125:,
120:1
114:U
103:1
97:F
71:,
66:2
60:U
54:,
49:1
43:U
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