Knowledge

Fσ set

Source 📝

652: 496: 263: 121: 521: 192: 162: 359: 230: 391: 324: 419: 693: 238: 630: 600: 717: 92: 686: 592: 616: 504: 175: 145: 722: 67: 679: 620: 586: 540: 406: 332: 209: 8: 410: 368: 402: 309: 32: 712: 667: 626: 596: 327: 45: 546: 394: 362: 165: 124: 40: 663: 582: 491:{\displaystyle A=\bigcup _{r\in \mathbb {Q} }\{(ry,y)\mid y\in \mathbb {R} \},} 706: 55: 532: 273: 36: 29: 75: 659: 277: 195: 651: 551: 588:
Real Analysis: Measure Theory, Integration, and Hilbert Spaces
523:
is the set of rational numbers, which is a countable set.
401:
set because it can be expressed as the union of all the
507: 422: 371: 335: 312: 241: 212: 178: 148: 95: 622:
Infinite Dimensional Analysis: A Hitchhiker's Guide
515: 490: 385: 353: 318: 258:{\displaystyle \mathbb {R} \setminus \mathbb {Q} } 257: 224: 186: 156: 115: 704: 615: 581: 687: 557:, any space having the property that every F 482: 447: 295:set, and the intersection of finitely many F 219: 213: 694: 680: 194:. More generally, any countable set in a 116:{\displaystyle \mathbf {\Sigma } _{2}^{0}} 577: 575: 509: 478: 441: 251: 243: 180: 150: 705: 572: 646: 13: 14: 734: 247: 650: 98: 609: 465: 450: 348: 336: 1: 565: 287:The union of countably many F 206:set, because every singleton 39:. The notation originated in 666:. You can help Knowledge by 516:{\displaystyle \mathbb {Q} } 187:{\displaystyle \mathbb {R} } 157:{\displaystyle \mathbb {Q} } 7: 526: 130: 10: 739: 645: 617:Aliprantis, Charalambos D. 593:Princeton University Press 585:; Shakarchi, Rami (2009), 265:of irrationals is not an F 625:, Springer, p. 138, 135:Each closed set is an F 718:Descriptive set theory 619:; Border, Kim (2006), 517: 492: 387: 355: 320: 259: 226: 188: 158: 117: 54: 44: 518: 493: 388: 356: 354:{\displaystyle (x,y)} 321: 260: 227: 225:{\displaystyle \{x\}} 189: 159: 118: 505: 420: 405:passing through the 369: 333: 310: 239: 210: 176: 146: 93: 53:: closed) and σ for 386:{\displaystyle x/y} 112: 16:In mathematics, an 513: 488: 446: 383: 351: 316: 255: 222: 184: 154: 113: 96: 675: 674: 429: 319:{\displaystyle A} 730: 696: 689: 682: 660:topology-related 654: 647: 637: 635: 613: 607: 605: 579: 522: 520: 519: 514: 512: 497: 495: 494: 489: 481: 445: 444: 392: 390: 389: 384: 379: 360: 358: 357: 352: 325: 323: 322: 317: 264: 262: 261: 256: 254: 246: 231: 229: 228: 223: 193: 191: 190: 185: 183: 163: 161: 160: 155: 153: 122: 120: 119: 114: 111: 106: 101: 738: 737: 733: 732: 731: 729: 728: 727: 703: 702: 701: 700: 643: 641: 640: 633: 614: 610: 603: 583:Stein, Elias M. 580: 573: 568: 560: 547:Borel hierarchy 536: 529: 508: 506: 503: 502: 477: 440: 433: 421: 418: 417: 400: 375: 370: 367: 366: 363:Cartesian plane 334: 331: 330: 311: 308: 307: 302: 298: 294: 290: 283: 268: 250: 242: 240: 237: 236: 211: 208: 207: 205: 199: 179: 177: 174: 173: 171: 149: 147: 144: 143: 138: 133: 125:Borel hierarchy 107: 102: 97: 94: 91: 90: 89:is the same as 88: 79: 73: 63:: sum, union). 21: 12: 11: 5: 736: 726: 725: 723:Topology stubs 720: 715: 699: 698: 691: 684: 676: 673: 672: 655: 639: 638: 631: 608: 601: 595:, p. 23, 570: 569: 567: 564: 563: 562: 558: 549: 544: 534: 528: 525: 511: 499: 498: 487: 484: 480: 476: 473: 470: 467: 464: 461: 458: 455: 452: 449: 443: 439: 436: 432: 428: 425: 409:with rational 398: 382: 378: 374: 350: 347: 344: 341: 338: 315: 300: 296: 292: 288: 281: 276:spaces, every 266: 253: 249: 245: 221: 218: 215: 203: 197: 182: 169: 152: 136: 132: 129: 110: 105: 100: 86: 77: 71: 19: 9: 6: 4: 3: 2: 735: 724: 721: 719: 716: 714: 711: 710: 708: 697: 692: 690: 685: 683: 678: 677: 671: 669: 665: 662:article is a 661: 656: 653: 649: 648: 644: 634: 632:9783540295877 628: 624: 623: 618: 612: 604: 602:9781400835560 598: 594: 590: 589: 584: 578: 576: 571: 561:set is closed 556: 554: 550: 548: 545: 542: 538: 531: 530: 524: 485: 474: 471: 468: 462: 459: 456: 453: 437: 434: 430: 426: 423: 416: 415: 414: 412: 408: 404: 396: 380: 376: 372: 364: 345: 342: 339: 329: 313: 304: 285: 279: 275: 270: 233: 216: 201: 167: 140: 128: 126: 108: 103: 83: 81: 69: 64: 62: 58: 57: 52: 48: 47: 42: 38: 34: 31: 27: 23: 668:expanding it 657: 642: 621: 611: 587: 552: 500: 305: 299:sets is an F 291:sets is an F 286: 271: 234: 141: 134: 84: 65: 60: 50: 25: 17: 15: 232:is closed. 43:with F for 37:closed sets 26:F-sigma set 707:Categories 566:References 365:such that 274:metrizable 68:complement 475:∈ 469:∣ 438:∈ 431:⋃ 248:∖ 166:rationals 99:Σ 74:set is a 30:countable 713:Topology 527:See also 395:rational 306:The set 278:open set 235:The set 142:The set 131:Examples 28:) is a 543:notion. 397:is an F 361:in the 326:of all 280:is an F 202:is an F 172:set in 168:is an F 123:in the 70:of an F 629:  599:  555:-space 539:— the 501:where 407:origin 328:points 284:set. 78:δ 61:French 51:French 41:French 24:(said 658:This 411:slope 403:lines 303:set. 269:set. 200:space 139:set. 56:somme 46:fermé 33:union 664:stub 627:ISBN 597:ISBN 541:dual 66:The 537:set 393:is 272:In 164:of 80:set 35:of 22:set 709:: 591:, 574:^ 413:: 127:. 82:. 695:e 688:t 681:v 670:. 636:. 606:. 559:σ 553:P 535:δ 533:G 510:Q 486:, 483:} 479:R 472:y 466:) 463:y 460:, 457:y 454:r 451:( 448:{ 442:Q 435:r 427:= 424:A 399:σ 381:y 377:/ 373:x 349:) 346:y 343:, 340:x 337:( 314:A 301:σ 297:σ 293:σ 289:σ 282:σ 267:σ 252:Q 244:R 220:} 217:x 214:{ 204:σ 198:1 196:T 181:R 170:σ 151:Q 137:σ 109:0 104:2 87:σ 85:F 76:G 72:σ 59:( 49:( 20:σ 18:F

Index

countable
union
closed sets
French
fermé
somme
complement
Gδ set
Borel hierarchy
rationals
T1 space
metrizable
open set
points
Cartesian plane
rational
lines
origin
slope
Gδ set
dual
Borel hierarchy
P-space


Stein, Elias M.
Real Analysis: Measure Theory, Integration, and Hilbert Spaces
Princeton University Press
ISBN
9781400835560

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