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P-space

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455: 289: 520: 496: 270:, Chapter a7 in Encyclopedia of General Topology, Edited by Klaas Pieter Hart, Jun-iti Nagata and Jerry E. Vaughan, 2003 Elsevier B.V. 225:(see the relevant entry for more information). Spaces satisfying the covering property introduced by Morita are sometimes also called 328: 429: 489: 424: 515: 187: 98: 195: 525: 482: 248: 61: 91: 191: 176: 157: 68: 378: 308: 222: 150: 106: 324: 57: 419: 398: 368: 298: 168: 356: 87: 466: 199: 167:
Different authors restrict their attention to topological spaces that satisfy various
509: 403: 161: 218: 359:; Henriksen, Melvin (September 1954), "Concerning Rings of Continuous Functions", 190:, in which arbitrary intersections of open sets are open. These in turn include 154: 146: 439: 110: 454: 95: 71:
of a different kind, i.e. non-topological spaces with additional structure.
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Misra, Arvind K. (December 1972), "A topological view of P-spaces",
373: 303: 284: 435: 164:, and a P-space is a space in which every point is a P-point. 319:
Hart, K.P. (2001). "P-point". In Hazewinkel, Michiel (ed.).
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Comparison of Topologies (Minimal and Maximal Topologies)
470: 105:is open. An equivalent condition is that countable 361:Transactions of the American Mathematical Society 290:Transactions of the American Mathematical Society 60:satisfying some given and previously introduced 175:-spaces in terms of their rings of continuous 171:. With the right axioms, one may characterize 490: 8: 321:Encyclopaedia of Mathematics, Supplement III 323:. Kluwer Academic Publishers. p. 297. 497: 483: 285:"Concerning rings of continuous functions" 402: 372: 343:Encyclopedia of General Topology, p. 278. 302: 259: 141:. Gillman and Henriksen also define a 94:is a topological space in which every 56:might be used generically to denote a 391:General Topology and Its Applications 7: 451: 449: 278: 276: 283:Gillman, L.; Henriksen, M. (1954). 14: 78:in the sense of Gillman–Henriksen 30:, there are various notions of a 521:Properties of topological spaces 453: 19:For the complexity class, see 1: 26:In the mathematical field of 469:. You can help Knowledge by 404:10.1016/0016-660X(72)90026-8 223:his (now solved) conjectures 221:in 1964, in connection with 113:are closed. In other words, 425:Encyclopedia of Mathematics 67:. This might apply also to 542: 448: 188:Alexandrov-discrete spaces 18: 213:A different notion of a 145:as a point at which any 247:has been introduced by 217:has been introduced by 129:are closed. The letter 266:Aisling E. McCluskey, 249:Alexander Arhangelskii 209:in the sense of Morita 192:locally finite spaces 177:real-valued functions 62:topological invariant 418:Hart, K.P. (2001) , 158:continuous functions 478: 477: 182:Special kinds of 169:separation axioms 58:topological space 16:Topological space 533: 516:General topology 499: 492: 485: 463:topology-related 457: 450: 432: 407: 406: 385: 376: 357:Gillman, Leonard 344: 341: 335: 334: 316: 306: 280: 271: 264: 194:, which include 186:-spaces include 133:stands for both 86:in the sense of 541: 540: 536: 535: 534: 532: 531: 530: 506: 505: 504: 503: 446: 417: 414: 388: 374:10.2307/1990875 355: 352: 350:Further reading 347: 342: 338: 331: 318: 304:10.2307/1990875 282: 281: 274: 265: 261: 257: 241: 231:normal P-spaces 227:Morita P-spaces 211: 200:discrete spaces 135:pseudo-discrete 126: 118: 80: 52:The expression 50: 24: 17: 12: 11: 5: 539: 537: 529: 528: 526:Topology stubs 523: 518: 508: 507: 502: 501: 494: 487: 479: 476: 475: 458: 444: 443: 433: 413: 412:External links 410: 409: 408: 397:(4): 349–362, 386: 367:(2): 340–362, 351: 348: 346: 345: 336: 329: 297:(2): 340–352. 272: 258: 256: 253: 243:A notion of a 240: 235: 210: 204: 124: 116: 79: 73: 49: 46: 15: 13: 10: 9: 6: 4: 3: 2: 538: 527: 524: 522: 519: 517: 514: 513: 511: 500: 495: 493: 488: 486: 481: 480: 474: 472: 468: 465:article is a 464: 459: 456: 452: 447: 441: 437: 434: 431: 427: 426: 421: 416: 415: 411: 405: 400: 396: 392: 387: 384: 380: 375: 370: 366: 362: 358: 354: 353: 349: 340: 337: 332: 330:1-4020-0198-3 326: 322: 314: 310: 305: 300: 296: 292: 291: 286: 279: 277: 273: 269: 263: 260: 254: 252: 250: 246: 239: 236: 234: 232: 228: 224: 220: 216: 208: 205: 203: 201: 197: 196:finite spaces 193: 189: 185: 180: 178: 174: 170: 165: 163: 159: 156: 152: 148: 144: 140: 136: 132: 128: 121:are open and 120: 112: 108: 104: 100: 97: 93: 89: 85: 77: 74: 72: 70: 66: 63: 59: 55: 47: 45: 43: 41: 36: 34: 29: 22: 471:expanding it 460: 445: 423: 394: 390: 364: 360: 339: 320: 294: 288: 267: 262: 244: 242: 237: 230: 226: 219:Kiiti Morita 214: 212: 206: 183: 181: 172: 166: 142: 138: 134: 130: 99:intersection 83: 81: 75: 64: 53: 51: 39: 38: 32: 31: 25: 155:real-valued 147:prime ideal 111:closed sets 48:Generic use 510:Categories 440:PlanetMath 317:Cited in 255:References 430:EMS Press 420:"P-space" 103:open sets 96:countable 92:Henriksen 37:and of a 238:p-spaces 207:P-spaces 76:P-spaces 28:topology 436:P-space 383:1990875 313:1990875 245:p-space 215:P-space 162:maximal 149:of the 143:P-point 88:Gillman 84:P-space 54:P-space 381:  327:  311:  107:unions 69:spaces 42:-space 35:-space 21:PSPACE 461:This 379:JSTOR 309:JSTOR 139:prime 467:stub 325:ISBN 198:and 151:ring 137:and 127:sets 119:sets 438:at 399:doi 369:doi 299:doi 229:or 160:is 153:of 109:of 101:of 512:: 428:, 422:, 393:, 377:, 365:77 363:, 307:. 295:77 293:. 287:. 275:^ 251:. 233:. 202:. 179:. 82:A 44:. 498:e 491:t 484:v 473:. 442:. 401:: 395:2 371:: 333:. 315:. 301:: 184:P 173:P 131:P 125:σ 123:F 117:δ 115:G 90:– 65:P 40:p 33:P 23:.

Index

PSPACE
topology
topological space
topological invariant
spaces
Gillman
Henriksen
countable
intersection
open sets
unions
closed sets
Gδ sets
Fσ sets
prime ideal
ring
real-valued
continuous functions
maximal
separation axioms
real-valued functions
Alexandrov-discrete spaces
locally finite spaces
finite spaces
discrete spaces
Kiiti Morita
his (now solved) conjectures
Alexander Arhangelskii

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