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σ-compact space

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71:. That terminology can be somewhat confusing as it does not fit the usual pattern of σ-(property) meaning a countable union of spaces satisfying (property); that's why such spaces are more commonly referred to explicitly as 179: 564: 554: 535: 519: 446: 154: 510: 76: 329: 559: 505: 68: 344: 201: 133: 125: 235:
is locally compact. This shows that for Hausdorff topological groups that are also Baire spaces,
148: 531: 515: 501: 442: 338: 262: 212: 101: 50: 32: 140:-compact nor locally compact. However, it is true that any locally compact Lindelöf space is 281: 109: 105: 270: 189: 113: 548: 311:
is second category (respectively Baire) if and only if the set of points at which is
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is locally compact everywhere. Therefore, the previous property tells us that if
193: 28: 332: – in analysis, a sequence of compact sets that converges on a given set 269:-compact. The converse, however, is not true; for example, the space of 231:-compact, Hausdorff topological group that is also a Baire space, then 384:"A question about local compactness and $ \sigma$ -compactness" 112:). The reverse implications do not hold, for example, standard 254:-compact, it would necessarily be locally compact since 315:
is locally compact is nonempty (respectively dense) in
292:-compact. However the product of an infinite number of 157: 334:
Pages displaying wikidata descriptions as a fallback
258:is a topological group that is also a Baire space. 174:{\displaystyle \mathbb {R} \setminus \mathbb {Q} } 173: 347: – Type of topological space in mathematics 242:The previous property implies for instance that 136:on any uncountable set is Lindelöf but neither 437:Hart, K.P.; Nagata, J.; Vaughan, J.E. (2004). 8: 167: 166: 159: 158: 156: 357: 239:-compactness implies local compactness. 163: 219:is locally compact at one point, then 514:, Holt, Rinehart and Winston (1970). 128:on the real line is Lindelöf but not 7: 75:, which is also equivalent to being 364:Steen, p. 19; Willard, p. 126. 124:-compact but not compact, and the 73:σ-compact (weakly) locally compact 25: 341: – Type of topological space 565:Properties of topological spaces 439:Encyclopedia of General Topology 296:-compact spaces may fail to be 273:, with the usual topology, is 1: 277:-compact but not hemicompact. 134:countable complement topology 511:Counterexamples in Topology 77:exhaustible by compact sets 581: 388:Mathematics Stack Exchange 330:Exhaustion by compact sets 555:Compactness (mathematics) 526:Willard, Stephen (2004). 441:. Elsevier. p. 170. 69:(weakly) locally compact 132:-compact. In fact, the 83:Properties and examples 530:. Dover Publications. 506:Seebach, J. Arthur Jr. 418:Steen, p. 75–76. 284:of a finite number of 204:at at least one point. 175: 56:A space is said to be 42:if it is the union of 345:Locally compact space 250:-compact: if it were 176: 18:Sigma-locally compact 155: 126:lower limit topology 96:-compact, and every 288:-compact spaces is 200:-compact, must be 171: 149:irrational numbers 100:-compact space is 263:hemicompact space 213:topological group 33:topological space 16:(Redirected from 572: 560:General topology 541: 528:General Topology 489: 488:Willard, p. 188. 486: 480: 479:Willard, p. 126. 477: 471: 470:Willard, p. 126. 468: 462: 461:Willard, p. 126. 459: 453: 452: 434: 428: 425: 419: 416: 410: 407: 401: 398: 392: 391: 380: 374: 371: 365: 362: 335: 180: 178: 177: 172: 170: 162: 108:has a countable 61:-locally compact 21: 580: 579: 575: 574: 573: 571: 570: 569: 545: 544: 538: 525: 498: 493: 492: 487: 483: 478: 474: 469: 465: 460: 456: 449: 436: 435: 431: 426: 422: 417: 413: 408: 404: 399: 395: 382: 381: 377: 372: 368: 363: 359: 354: 333: 326: 307:-compact space 202:locally compact 153: 152: 114:Euclidean space 85: 23: 22: 15: 12: 11: 5: 578: 576: 568: 567: 562: 557: 547: 546: 543: 542: 536: 523: 502:Steen, Lynn A. 497: 494: 491: 490: 481: 472: 463: 454: 447: 429: 420: 411: 402: 393: 375: 366: 356: 355: 353: 350: 349: 348: 342: 339:Lindelöf space 336: 325: 322: 321: 320: 301: 278: 259: 240: 205: 186: 169: 165: 161: 145: 84: 81: 63:if it is both 35:is said to be 24: 14: 13: 10: 9: 6: 4: 3: 2: 577: 566: 563: 561: 558: 556: 553: 552: 550: 539: 537:0-486-43479-6 533: 529: 524: 521: 520:0-03-079485-4 517: 513: 512: 507: 503: 500: 499: 495: 485: 482: 476: 473: 467: 464: 458: 455: 450: 448:0 444 50355 2 444: 440: 433: 430: 427:Steen, p. 50. 424: 421: 415: 412: 409:Steen, p. 56. 406: 403: 400:Steen, p. 19. 397: 394: 389: 385: 379: 376: 373:Steen, p. 21. 370: 367: 361: 358: 351: 346: 343: 340: 337: 331: 328: 327: 323: 318: 314: 310: 306: 302: 299: 295: 291: 287: 283: 279: 276: 272: 268: 264: 260: 257: 253: 249: 245: 241: 238: 234: 230: 226: 222: 218: 214: 210: 206: 203: 199: 196:that is also 195: 191: 187: 184: 150: 146: 143: 139: 135: 131: 127: 123: 119: 115: 111: 107: 103: 99: 95: 91: 90:compact space 87: 86: 82: 80: 78: 74: 70: 67:-compact and 66: 62: 60: 54: 52: 49: 45: 41: 39: 34: 30: 19: 527: 509: 484: 475: 466: 457: 438: 432: 423: 414: 405: 396: 387: 378: 369: 360: 316: 312: 308: 304: 297: 293: 289: 285: 274: 266: 255: 251: 247: 243: 236: 232: 228: 224: 220: 216: 208: 197: 182: 141: 137: 129: 121: 117: 104:(i.e. every 97: 93: 72: 64: 58: 57: 55: 37: 36: 26: 194:Baire space 29:mathematics 549:Categories 496:References 106:open cover 300:-compact. 271:rationals 190:Hausdorff 185:-compact. 164:∖ 144:-compact. 51:subspaces 44:countably 324:See also 110:subcover 102:Lindelöf 40:-compact 282:product 246:is not 181:is not 48:compact 534:  518:  445:  261:Every 88:Every 352:Notes 227:is a 211:is a 147:(The 120:) is 46:many 532:ISBN 516:ISBN 504:and 443:ISBN 280:The 215:and 31:, a 265:is 207:If 92:is 27:In 551:: 508:; 386:. 303:A 192:, 188:A 151:) 79:. 53:. 540:. 522:. 451:. 390:. 319:. 317:X 313:X 309:X 305:σ 298:σ 294:σ 290:σ 286:σ 275:σ 267:σ 256:R 252:σ 248:σ 244:R 237:σ 233:G 229:σ 225:G 221:G 217:G 209:G 198:σ 183:σ 168:Q 160:R 142:σ 138:σ 130:σ 122:σ 118:R 116:( 98:σ 94:σ 65:σ 59:σ 38:σ 20:)

Index

Sigma-locally compact
mathematics
topological space
countably
compact
subspaces
(weakly) locally compact
exhaustible by compact sets
compact space
Lindelöf
open cover
subcover
Euclidean space
lower limit topology
countable complement topology
irrational numbers
Hausdorff
Baire space
locally compact
topological group
hemicompact space
rationals
product
Exhaustion by compact sets
Lindelöf space
Locally compact space
"A question about local compactness and $ \sigma$ -compactness"
ISBN
0 444 50355 2
Steen, Lynn A.

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