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Countably generated space

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An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of
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https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
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of countably generated spaces is countably generated. Therefore, the countably generated spaces form a
1203: 83: 1210:− Tightness is a cardinal function related to countably generated spaces and their generalizations. 199: 133: 310: 21: 244: 1170: 793: 394: 76: 72: 1147: 336: 1274: 1221: 865: 765: 466: 366: 290: 270: 224: 204: 181: 161: 105: 50: 26: 1317: 17: 1192: â€“ topology in which the intersection of any family of open sets is open 1262: 68: 1161:
Any subspace of a countably generated space is again countably generated.
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of a countably generated space is again countably generated. Similarly, a
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The countably generated spaces are precisely the spaces having countable
1123:{\displaystyle x\in {\overline {\left\{x_{1},x_{2},\ldots \right\}}}.} 652:{\displaystyle B_{1}\subseteq A_{1},B_{2}\subseteq A_{2},\ldots } 1158:. They are the coreflective hull of all countable spaces. 333:
equals the union of closures of all countable subsets of
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is countably generated if and only if the closure of any
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Pages displaying short descriptions of redirect targets
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Pages displaying wikidata descriptions as a fallback
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Berlin: 7: 1259: 1257: 1134:has strong countable fan tightness. 898: 855:{\displaystyle A_{1},A_{2},\ldots } 678: 499: 456:{\displaystyle A_{1},A_{2},\ldots } 14: 1261: 1156:category of topological spaces 786:countable strong fan tightness 1: 152:{\displaystyle V\subseteq X,} 1277:. You can help Knowledge by 1132:strong FrĂ©chet–Urysohn space 1112: 962: 942: 922: 742: 701: 563: 543: 523: 326:{\displaystyle A\subseteq X} 198:whenever for each countable 1345: 1256: 1173:) is countably generated. 1190:Finitely generated space 1152:coreflective subcategory 862:of subsets of the space 463:of subsets of the space 387:countable fan tightness 359:Countable fan tightness 260:{\displaystyle V\cap U} 1124: 1052: 980: 876: 856: 810: 809:{\displaystyle x\in X} 776: 754: 653: 581: 477: 457: 411: 410:{\displaystyle x\in X} 377: 350: 327: 301: 281: 261: 235: 215: 192: 172: 153: 116: 61: 37: 1199:Locally closed subset 1125: 1053: 981: 877: 857: 811: 777: 755: 654: 587:there are finite set 582: 478: 458: 412: 378: 351: 328: 302: 282: 262: 236: 216: 193: 173: 154: 117: 67:is determined by the 62: 38: 1204:Tightness (topology) 1062: 990: 886: 866: 820: 794: 766: 762:A topological space 663: 591: 487: 467: 421: 395: 367: 363:A topological space 337: 311: 291: 271: 245: 225: 205: 182: 162: 134: 130:if for every subset 106: 102:A topological space 86:—therefore the name 51: 27: 816:and every sequence 790:if for every point 417:and every sequence 391:if for every point 126:countably generated 47:if the topology of 45:countably generated 1120: 1048: 976: 907: 906: 872: 852: 806: 772: 750: 687: 686: 649: 577: 508: 507: 473: 453: 407: 373: 349:{\displaystyle A.} 346: 323: 297: 277: 257: 231: 211: 188: 168: 149: 112: 57: 33: 1286: 1285: 1115: 986:there are points 965: 945: 925: 897: 875:{\displaystyle X} 775:{\displaystyle X} 745: 704: 677: 566: 546: 526: 498: 476:{\displaystyle X} 376:{\displaystyle X} 300:{\displaystyle X} 280:{\displaystyle U} 234:{\displaystyle X} 214:{\displaystyle U} 191:{\displaystyle X} 171:{\displaystyle V} 115:{\displaystyle X} 94:is used as well. 60:{\displaystyle X} 36:{\displaystyle X} 22:topological space 1336: 1324:General topology 1307: 1300: 1293: 1271:topology-related 1265: 1258: 1233: 1209: 1195: 1178:Arens–Fort space 1171:metrizable space 1129: 1127: 1126: 1121: 1116: 1111: 1107: 1100: 1099: 1087: 1086: 1072: 1057: 1055: 1054: 1049: 1041: 1040: 1028: 1027: 1015: 1014: 1002: 1001: 985: 983: 982: 977: 966: 961: 960: 951: 946: 941: 940: 931: 926: 921: 920: 911: 908: 905: 881: 879: 878: 873: 861: 859: 858: 853: 845: 844: 832: 831: 815: 813: 812: 807: 788: 787: 781: 779: 778: 773: 759: 757: 756: 751: 746: 741: 734: 733: 721: 720: 710: 705: 700: 699: 698: 688: 685: 673: 658: 656: 655: 650: 642: 641: 629: 628: 616: 615: 603: 602: 586: 584: 583: 578: 567: 562: 561: 552: 547: 542: 541: 532: 527: 522: 521: 512: 509: 506: 482: 480: 479: 474: 462: 460: 459: 454: 446: 445: 433: 432: 416: 414: 413: 408: 389: 388: 382: 380: 379: 374: 355: 353: 352: 347: 332: 330: 329: 324: 306: 304: 303: 298: 287:. Equivalently, 286: 284: 283: 278: 266: 264: 263: 258: 240: 238: 237: 232: 220: 218: 217: 212: 197: 195: 194: 189: 177: 175: 174: 169: 158: 156: 155: 150: 128: 127: 121: 119: 118: 113: 92: 91: 73:sequential space 66: 64: 63: 58: 42: 40: 39: 34: 1344: 1343: 1339: 1338: 1337: 1335: 1334: 1333: 1314: 1313: 1312: 1311: 1254: 1240: 1222:Herrlich, Horst 1220: 1217: 1207: 1193: 1186: 1167: 1148:topological sum 1140: 1091: 1078: 1077: 1073: 1060: 1059: 1032: 1019: 1006: 993: 988: 987: 952: 932: 912: 884: 883: 864: 863: 836: 823: 818: 817: 792: 791: 785: 784: 764: 763: 725: 712: 711: 690: 674: 661: 660: 633: 620: 607: 594: 589: 588: 553: 533: 513: 485: 484: 465: 464: 437: 424: 419: 418: 393: 392: 386: 385: 365: 364: 361: 335: 334: 309: 308: 289: 288: 269: 268: 243: 242: 223: 222: 203: 202: 180: 179: 160: 159: 132: 131: 125: 124: 104: 103: 100: 90:countably tight 89: 88: 49: 48: 25: 24: 12: 11: 5: 1342: 1340: 1332: 1331: 1329:Topology stubs 1326: 1316: 1315: 1310: 1309: 1302: 1295: 1287: 1284: 1283: 1266: 1252: 1251: 1246: 1239: 1238:External links 1236: 1235: 1234: 1216: 1213: 1212: 1211: 1201: 1196: 1185: 1182: 1166: 1163: 1139: 1136: 1119: 1114: 1110: 1106: 1103: 1098: 1094: 1090: 1085: 1081: 1076: 1070: 1067: 1047: 1044: 1039: 1035: 1031: 1026: 1022: 1018: 1013: 1009: 1005: 1000: 996: 975: 972: 969: 964: 959: 955: 949: 944: 939: 935: 929: 924: 919: 915: 904: 900: 894: 891: 871: 851: 848: 843: 839: 835: 830: 826: 805: 802: 799: 771: 749: 744: 740: 737: 732: 728: 724: 719: 715: 708: 703: 697: 693: 684: 680: 671: 668: 648: 645: 640: 636: 632: 627: 623: 619: 614: 610: 606: 601: 597: 576: 573: 570: 565: 560: 556: 550: 545: 540: 536: 530: 525: 520: 516: 505: 501: 495: 492: 472: 452: 449: 444: 440: 436: 431: 427: 406: 403: 400: 372: 360: 357: 345: 342: 322: 319: 316: 296: 276: 256: 253: 250: 230: 210: 187: 167: 148: 145: 142: 139: 111: 99: 96: 56: 32: 13: 10: 9: 6: 4: 3: 2: 1341: 1330: 1327: 1325: 1322: 1321: 1319: 1308: 1303: 1301: 1296: 1294: 1289: 1288: 1282: 1280: 1276: 1273:article is a 1272: 1267: 1264: 1260: 1255: 1250: 1247: 1245: 1242: 1241: 1237: 1231: 1227: 1223: 1219: 1218: 1214: 1205: 1202: 1200: 1197: 1191: 1188: 1187: 1183: 1181: 1179: 1174: 1172: 1164: 1162: 1159: 1157: 1153: 1149: 1145: 1137: 1135: 1133: 1117: 1108: 1104: 1101: 1096: 1092: 1088: 1083: 1079: 1074: 1068: 1065: 1045: 1042: 1037: 1033: 1029: 1024: 1020: 1016: 1011: 1007: 1003: 998: 994: 973: 970: 967: 957: 953: 947: 937: 933: 927: 917: 913: 902: 892: 889: 869: 849: 846: 841: 837: 833: 828: 824: 803: 800: 797: 789: 769: 760: 747: 738: 735: 730: 726: 722: 717: 713: 706: 695: 691: 682: 669: 666: 646: 643: 638: 634: 630: 625: 621: 617: 612: 608: 604: 599: 595: 574: 571: 568: 558: 554: 548: 538: 534: 528: 518: 514: 503: 493: 490: 470: 450: 447: 442: 438: 434: 429: 425: 404: 401: 398: 390: 370: 358: 356: 343: 340: 320: 317: 314: 294: 274: 267:is closed in 254: 251: 248: 228: 208: 201: 185: 178:is closed in 165: 146: 143: 140: 137: 129: 109: 97: 95: 93: 85: 80: 78: 77:FrĂ©chet space 74: 70: 54: 46: 30: 23: 19: 1279:expanding it 1268: 1253: 1225: 1175: 1168: 1160: 1141: 783: 761: 384: 362: 123: 101: 87: 81: 44: 15: 18:mathematics 1318:Categories 1215:References 1138:Properties 1058:such that 882:such that 659:such that 483:such that 122:is called 98:Definition 43:is called 1113:¯ 1105:… 1069:∈ 1046:… 1030:∈ 1004:∈ 971:⋯ 968:∩ 963:¯ 948:∩ 943:¯ 923:¯ 899:⋂ 893:∈ 850:… 801:∈ 743:¯ 739:⋯ 736:∪ 723:∪ 702:¯ 679:⋃ 670:∈ 647:… 631:⊆ 605:⊆ 572:⋯ 569:∩ 564:¯ 549:∩ 544:¯ 524:¯ 500:⋂ 494:∈ 451:… 402:∈ 318:⊆ 252:∩ 141:⊆ 84:tightness 69:countable 1230:Springer 1224:(1968). 1184:See also 1165:Examples 1144:quotient 241:the set 200:subspace 1154:of the 1130:Every 75:(or a 1269:This 1275:stub 782:has 383:has 20:, a 221:of 16:In 1320:: 1180:. 1142:A 1306:e 1299:t 1292:v 1281:. 1232:. 1118:. 1109:} 1102:, 1097:2 1093:x 1089:, 1084:1 1080:x 1075:{ 1066:x 1043:, 1038:2 1034:A 1025:2 1021:x 1017:, 1012:1 1008:A 999:1 995:x 974:, 958:2 954:A 938:1 934:A 928:= 918:n 914:A 903:n 890:x 870:X 847:, 842:2 838:A 834:, 829:1 825:A 804:X 798:x 770:X 748:. 731:2 727:B 718:1 714:B 707:= 696:n 692:B 683:n 667:x 644:, 639:2 635:A 626:2 622:B 618:, 613:1 609:A 600:1 596:B 575:, 559:2 555:A 539:1 535:A 529:= 519:n 515:A 504:n 491:x 471:X 448:, 443:2 439:A 435:, 430:1 426:A 405:X 399:x 371:X 344:. 341:A 321:X 315:A 295:X 275:U 255:U 249:V 229:X 209:U 186:X 166:V 147:, 144:X 138:V 110:X 55:X 31:X

Index

mathematics
topological space
countable
sequential space
Fréchet space
tightness
subspace
strong FrĂ©chet–Urysohn space
quotient
topological sum
coreflective subcategory
category of topological spaces
metrizable space
Arens–Fort space
Finitely generated space
Locally closed subset
Tightness (topology)
Herrlich, Horst
Springer

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