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Most authors do not require that neighborhoods be open sets because writing "open" in front of "neighborhood" when this property is needed is not overly onerous and because requiring that they always be open would also greatly limit the usefulness of terms such as "closed neighborhood" and "compact
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This is because, by assumption, vector addition is separately continuous in the induced topology. Therefore, the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the space is a
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In any topological space, the neighbourhood system for a point is also a neighbourhood basis for the point. The set of all open neighbourhoods at a point forms a neighbourhood basis at that point. For any point
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have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods." Similarly, a neighbourhood that is also a
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and it will be clearly indicated if it instead refers to a neighborhood of a set. So for instance, a statement such as "a neighbourhood in
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1149:{\displaystyle {\mathcal {N}}(x)=\left\{V\subseteq X~:~B\subseteq V{\text{ for some }}B\in {\mathcal {B}}\right\}\!\!\;.}
572:, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.
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2942:{\displaystyle \left\{\mu \in {\mathcal {M}}(E):\left|\mu f_{i}-\nu f_{i}\right|<r_{i},\,i=1,\dots ,n\right\}}
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568:, although many times, these neighbourhoods are not necessarily open.
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2667:{\displaystyle {\mathcal {B}}=\left\{B_{1/n}:n=1,2,3,\dots \right\}}
3655: – A vector space with a topology defined by convex open sets
3598:{\displaystyle \left(x_{N}\right)_{N\in {\mathcal {N}}}\to u}
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1825:{\displaystyle (-2,2),\;,\;\cup \mathbb {Q} ,\;\mathbb {R} }
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For example, all of the following sets are neighborhoods of
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of the neighbourhood filter; this means that it is a subset
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3875:. Berlin New York: Springer Science & Business Media.
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3637: – Collection of open sets used to define a topology
1289:{\displaystyle \left({\mathcal {N}}(x),\supseteq \right)}
1192:{\displaystyle {\mathcal {B}}\subseteq {\mathcal {N}}(x)}
794:{\displaystyle {\mathcal {B}}\subseteq {\mathcal {N}}(x)}
3445:{\displaystyle \left(x_{N}\right)_{N\in {\mathcal {N}}}}
3667: – Collection of subsets that generate a topology
3128:{\displaystyle {\mathcal {N}}(x)={\mathcal {N}}(0)+x.}
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a neighborhood of any other point. Said differently,
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but none of the following sets are neighborhoods of
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Pages displaying wikidata descriptions as a fallback
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3643: – Family of sets representing "large" sets
1433:such that the collection of all possible finite
942:in the neighbourhood basis that is contained in
2494:{\displaystyle x\in \operatorname {int} _{X}N.}
663:is the same as the neighbourhood filter of the
2382:{\displaystyle x\in \operatorname {int} _{X}N}
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3050:Seminormed spaces and topological groups
1319:(importantly, this partial order is the
1011:if and only if the neighbourhood filter
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3653:Locally convex topological vector space
3501:
3462:
2763:{\displaystyle {\mathcal {N}}(x)=\{X\}}
2705:the neighbourhood system for any point
1964:
1558:{\displaystyle N\subseteq \mathbb {R} }
836:{\displaystyle V\in {\mathcal {N}}(x),}
3774:(Third ed.). Dover. p. 41.
3510:{\displaystyle x_{N}\in N\setminus U}
637:The neighbourhood filter for a point
7:
2777:on the space of measures on a space
520:Importantly, a "neighbourhood" does
3540:{\displaystyle N\in {\mathcal {N}}}
2674:. This means every metric space is
866:{\displaystyle B\in {\mathcal {B}}}
473:. Equivalently, a neighborhood of
3039:{\displaystyle r_{1},\dots ,r_{n}}
1743:
1631:{\displaystyle (-r,r)\subseteq N.}
25:
3179:{\displaystyle u\in U\subseteq X}
466:{\displaystyle x\in U\subseteq N}
72:{\displaystyle {\mathcal {N}}(x)}
41:complete system of neighbourhoods
3142:or the topology is defined by a
3373:if and only if there exists an
2725:only contains the whole space,
1461:forms a neighbourhood basis at
902:That is, for any neighbourhood
138:Neighbourhood of a point or set
3862:General Topology: Chapters 1–4
3589:
3390:{\displaystyle {\mathcal {N}}}
3270:{\displaystyle {\mathcal {N}}}
3203:{\displaystyle {\mathcal {N}}}
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1454:{\displaystyle {\mathcal {S}}}
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1052:{\displaystyle {\mathcal {B}}}
1028:{\displaystyle {\mathcal {N}}}
984:{\displaystyle {\mathcal {B}}}
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421:there exists some open subset
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1:
3830:. Addison-Wesley Publishing.
3302:{\displaystyle \,\supseteq .}
3210:be a neighbourhood basis for
2433:is a neighborhood of a point
1199:is a neighbourhood basis for
895:{\displaystyle B\subseteq V.}
3471:{\displaystyle X\setminus U}
2223:{\displaystyle N\subseteq X}
2056:{\displaystyle \mathbb {Q} }
1673:{\displaystyle \mathbb {R} }
1565:for which there exists some
1506:{\displaystyle \mathbb {R} }
922:we can find a neighbourhood
327:{\displaystyle N\subseteq X}
3659:Neighbourhood (mathematics)
3046:are positive real numbers.
2800:a neighbourhood base about
3995:
3932:Willard, Stephen (2004) .
3842:(See Chapter 2, Section 4)
1517:then the neighborhoods of
1312:{\displaystyle \supseteq }
3822:Willard, Stephen (1970).
3770:Mendelson, Bert (1990) .
3285:it by superset inclusion
493:is any set that contains
103:is the collection of all
3873:Éléments de mathématique
3772:Introduction to Topology
3000:to the real numbers and
536:, etc.) set is called a
2980:bounded functions from
2135:{\displaystyle u\in U,}
2090:is an open subset of a
1410:each of which contains
556:connected neighbourhood
174:in a topological space
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1537:are all those subsets
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1339:neighbourhood subbasis
1331:Neighbourhood subbasis
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688:{\displaystyle \{x\}.}
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656:{\displaystyle x\in X}
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570:Locally compact spaces
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548:compact neighbourhood
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31:and related areas of
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3671:Tubular neighborhood
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3547:(which implies that
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2813:{\displaystyle \nu }
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2589:neighbourhood basis
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2265:topological interior
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1296:with respect to the
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1119: for some
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991:is a local basis at
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611:neighbourhood filter
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576:Neighbourhood filter
540:closed neighbourhood
515:topological interior
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45:neighbourhood filter
37:neighbourhood system
3647:Filters in topology
3641:Filter (set theory)
2703:indiscrete topology
2578:{\displaystyle 1/n}
2504:Neighbourhood bases
2204:More generally, if
706:neighbourhood basis
698:Neighbourhood basis
566:neighbourhood basis
562:functional analysis
18:Neighborhood filter
3944:Dover Publications
3868:Topologie Générale
3737:{\displaystyle X.}
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3333:a neighborhood of
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3283:partially ordering
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3246:{\displaystyle X.}
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2793:{\displaystyle E,}
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2533:, the sequence of
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2303:{\displaystyle X,}
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2197:{\displaystyle X.}
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1515:Euclidean topology
1503:
1477:{\displaystyle x.}
1474:
1451:
1426:{\displaystyle x,}
1423:
1403:{\displaystyle X,}
1400:
1377:
1353:
1309:
1286:
1233:
1209:
1189:
1146:
1049:
1025:
1001:
981:
958:{\displaystyle V.}
955:
932:
912:
892:
863:
843:there exists some
833:
801:such that for all
791:
744:
722:neighbourhood base
685:
653:
629:{\displaystyle x.}
626:
594:
503:
483:
463:
431:
408:
388:
368:
348:
324:
297:
277:
253:{\displaystyle x.}
250:
227:
207:
184:
164:
146:open neighbourhood
123:{\displaystyle x.}
120:
89:
69:
3953:978-0-486-43479-7
3916:978-0-387-90972-1
3882:978-3-540-64241-1
3857:Bourbaki, Nicolas
3812:, pp. 31–37.
3797:, pp. 17–21.
3710:{\displaystyle X}
3618:{\displaystyle X}
3366:{\displaystyle X}
3346:{\displaystyle u}
3322:{\displaystyle U}
3223:{\displaystyle u}
3140:topological group
2993:{\displaystyle E}
2718:{\displaystyle x}
2694:{\displaystyle X}
2550:{\displaystyle x}
2522:{\displaystyle x}
2426:{\displaystyle N}
2402:{\displaystyle N}
2350:) of every point
2343:{\displaystyle X}
2323:{\displaystyle N}
2280:{\displaystyle N}
2174:{\displaystyle u}
2154:{\displaystyle U}
2106:{\displaystyle X}
2092:topological space
2083:{\displaystyle U}
2017:
2002:
1987:
1845:{\displaystyle 0}
1651:{\displaystyle 0}
1530:{\displaystyle 0}
1356:{\displaystyle x}
1212:{\displaystyle x}
1120:
1107:
1101:
1004:{\displaystyle x}
935:{\displaystyle B}
915:{\displaystyle V}
747:{\displaystyle x}
597:{\displaystyle x}
506:{\displaystyle x}
486:{\displaystyle x}
434:{\displaystyle U}
411:{\displaystyle X}
391:{\displaystyle x}
371:{\displaystyle N}
351:{\displaystyle x}
300:{\displaystyle X}
280:{\displaystyle x}
230:{\displaystyle X}
210:{\displaystyle U}
187:{\displaystyle X}
167:{\displaystyle x}
101:topological space
92:{\displaystyle x}
16:(Redirected from
3986:
3979:General topology
3965:
3935:General Topology
3928:
3903:General Topology
3899:Dixmier, Jacques
3894:
3843:
3841:
3829:
3826:General Topology
3819:
3813:
3807:
3798:
3792:
3786:
3785:
3767:
3755:
3751:
3745:
3743:
3741:
3740:
3735:
3716:
3714:
3713:
3708:
3691:
3676:
3624:
3622:
3621:
3616:
3604:
3602:
3601:
3596:
3588:
3587:
3586:
3585:
3572:
3568:
3567:
3546:
3544:
3543:
3538:
3536:
3535:
3516:
3514:
3513:
3508:
3494:
3493:
3477:
3475:
3474:
3469:
3451:
3449:
3448:
3443:
3441:
3440:
3439:
3438:
3425:
3421:
3420:
3396:
3394:
3393:
3388:
3386:
3385:
3372:
3370:
3369:
3364:
3352:
3350:
3349:
3344:
3328:
3326:
3325:
3320:
3308:
3306:
3305:
3300:
3276:
3274:
3273:
3268:
3266:
3265:
3252:
3250:
3249:
3244:
3229:
3227:
3226:
3221:
3209:
3207:
3206:
3201:
3199:
3198:
3185:
3183:
3182:
3177:
3134:
3132:
3131:
3126:
3106:
3105:
3087:
3086:
3056:seminormed space
3045:
3043:
3042:
3037:
3035:
3034:
3016:
3015:
2999:
2997:
2996:
2991:
2975:
2973:
2972:
2967:
2965:
2964:
2948:
2946:
2945:
2940:
2938:
2934:
2908:
2907:
2895:
2891:
2890:
2889:
2874:
2873:
2844:
2843:
2819:
2817:
2816:
2811:
2799:
2797:
2796:
2791:
2769:
2767:
2766:
2761:
2738:
2737:
2724:
2722:
2721:
2716:
2700:
2698:
2697:
2692:
2673:
2671:
2670:
2665:
2663:
2659:
2628:
2627:
2623:
2602:
2601:
2584:
2582:
2581:
2576:
2571:
2556:
2554:
2553:
2548:
2528:
2526:
2525:
2520:
2500:
2498:
2497:
2492:
2481:
2480:
2458:
2456:
2455:
2450:
2432:
2430:
2429:
2424:
2408:
2406:
2405:
2400:
2388:
2386:
2385:
2380:
2372:
2371:
2349:
2347:
2346:
2341:
2329:
2327:
2326:
2321:
2309:
2307:
2306:
2301:
2286:
2284:
2283:
2278:
2262:
2260:
2259:
2254:
2246:
2245:
2229:
2227:
2226:
2221:
2203:
2201:
2200:
2195:
2180:
2178:
2177:
2172:
2160:
2158:
2157:
2152:
2141:
2139:
2138:
2133:
2112:
2110:
2109:
2104:
2089:
2087:
2086:
2081:
2065:rational numbers
2062:
2060:
2059:
2054:
2052:
2040:
2038:
2037:
2032:
2030:
2026:
2019:
2010:
2004:
1995:
1989:
1980:
1941:
1876:
1851:
1849:
1848:
1843:
1831:
1829:
1828:
1823:
1821:
1812:
1679:
1677:
1676:
1671:
1669:
1657:
1655:
1654:
1649:
1637:
1635:
1634:
1629:
1593:
1591:
1590:
1585:
1564:
1562:
1561:
1556:
1554:
1536:
1534:
1533:
1528:
1512:
1510:
1509:
1504:
1502:
1483:
1481:
1480:
1475:
1460:
1458:
1457:
1452:
1450:
1449:
1432:
1430:
1429:
1424:
1409:
1407:
1406:
1401:
1386:
1384:
1383:
1378:
1376:
1375:
1362:
1360:
1359:
1354:
1341:
1340:
1318:
1316:
1315:
1310:
1295:
1293:
1292:
1287:
1285:
1281:
1265:
1264:
1242:
1240:
1239:
1234:
1232:
1231:
1218:
1216:
1215:
1210:
1198:
1196:
1195:
1190:
1179:
1178:
1169:
1168:
1155:
1153:
1152:
1147:
1139:
1135:
1134:
1133:
1121:
1118:
1105:
1099:
1072:
1071:
1058:
1056:
1055:
1050:
1048:
1047:
1034:
1032:
1031:
1026:
1024:
1023:
1010:
1008:
1007:
1002:
990:
988:
987:
982:
980:
979:
964:
962:
961:
956:
941:
939:
938:
933:
921:
919:
918:
913:
901:
899:
898:
893:
872:
870:
869:
864:
862:
861:
842:
840:
839:
834:
820:
819:
800:
798:
797:
792:
781:
780:
771:
770:
753:
751:
750:
745:
732:
731:
724:
723:
716:
715:
708:
707:
694:
692:
691:
686:
662:
660:
659:
654:
635:
633:
632:
627:
603:
601:
600:
595:
558:
557:
550:
549:
542:
541:
512:
510:
509:
504:
492:
490:
489:
484:
472:
470:
469:
464:
440:
438:
437:
432:
417:
415:
414:
409:
397:
395:
394:
389:
377:
375:
374:
369:
357:
355:
354:
349:
333:
331:
330:
325:
306:
304:
303:
298:
286:
284:
283:
278:
266:
265:
259:
257:
256:
251:
236:
234:
233:
228:
216:
214:
213:
208:
193:
191:
190:
185:
173:
171:
170:
165:
148:
147:
129:
127:
126:
121:
98:
96:
95:
90:
78:
76:
75:
70:
59:
58:
21:
3994:
3993:
3989:
3988:
3987:
3985:
3984:
3983:
3969:
3968:
3954:
3931:
3917:
3907:Springer-Verlag
3897:
3883:
3855:
3852:
3847:
3846:
3838:
3821:
3820:
3816:
3808:
3801:
3793:
3789:
3782:
3769:
3768:
3764:
3759:
3758:
3752:
3748:
3723:
3722:
3699:
3698:
3692:
3688:
3683:
3674:
3635:Base (topology)
3631:
3607:
3606:
3559:
3555:
3554:
3549:
3548:
3519:
3518:
3485:
3480:
3479:
3454:
3453:
3412:
3408:
3407:
3402:
3401:
3375:
3374:
3355:
3354:
3335:
3334:
3311:
3310:
3287:
3286:
3255:
3254:
3232:
3231:
3212:
3211:
3188:
3187:
3156:
3155:
3152:
3076:
3075:
3026:
3007:
3002:
3001:
2982:
2981:
2956:
2951:
2950:
2899:
2881:
2865:
2861:
2857:
2831:
2827:
2822:
2821:
2802:
2801:
2779:
2778:
2727:
2726:
2707:
2706:
2683:
2682:
2676:first-countable
2611:
2610:
2606:
2591:
2590:
2559:
2558:
2539:
2538:
2511:
2510:
2472:
2461:
2460:
2459:if and only if
2435:
2434:
2415:
2414:
2391:
2390:
2363:
2352:
2351:
2332:
2331:
2312:
2311:
2289:
2288:
2269:
2268:
2237:
2232:
2231:
2230:is any set and
2206:
2205:
2183:
2182:
2163:
2162:
2143:
2142:
2115:
2114:
2113:then for every
2095:
2094:
2072:
2071:
2043:
2042:
1971:
1967:
1854:
1853:
1834:
1833:
1682:
1681:
1660:
1659:
1640:
1639:
1596:
1595:
1570:
1569:
1539:
1538:
1519:
1518:
1493:
1492:
1489:
1463:
1462:
1439:
1438:
1437:of elements of
1412:
1411:
1389:
1388:
1365:
1364:
1345:
1344:
1338:
1337:
1333:
1301:
1300:
1258:
1254:
1249:
1248:
1221:
1220:
1219:if and only if
1201:
1200:
1158:
1157:
1089:
1085:
1061:
1060:
1037:
1036:
1013:
1012:
993:
992:
969:
968:
944:
943:
924:
923:
904:
903:
875:
874:
845:
844:
803:
802:
760:
759:
736:
735:
729:
728:
721:
720:
713:
712:
705:
704:
700:
668:
667:
639:
638:
615:
614:
586:
585:
555:
554:
547:
546:
544:(respectively,
539:
538:
528:(respectively,
495:
494:
475:
474:
443:
442:
423:
422:
400:
399:
380:
379:
360:
359:
358:; explicitly,
340:
339:
310:
309:
289:
288:
269:
268:
263:
262:
239:
238:
219:
218:
199:
198:
176:
175:
156:
155:
150:of a point (or
145:
144:
135:
109:
108:
81:
80:
48:
47:
23:
22:
15:
12:
11:
5:
3992:
3990:
3982:
3981:
3971:
3970:
3967:
3966:
3952:
3929:
3915:
3895:
3881:
3851:
3848:
3845:
3844:
3836:
3814:
3799:
3787:
3780:
3761:
3760:
3757:
3756:
3754:neighborhood".
3746:
3733:
3730:
3720:
3706:
3696:
3685:
3684:
3682:
3679:
3678:
3677:
3668:
3662:
3656:
3650:
3644:
3638:
3630:
3627:
3614:
3594:
3591:
3584:
3579:
3576:
3571:
3566:
3562:
3558:
3534:
3529:
3526:
3506:
3503:
3500:
3497:
3492:
3488:
3467:
3464:
3461:
3437:
3432:
3429:
3424:
3419:
3415:
3411:
3384:
3362:
3342:
3332:
3318:
3298:
3295:
3264:
3242:
3239:
3219:
3197:
3175:
3172:
3169:
3166:
3163:
3151:
3148:
3124:
3121:
3118:
3115:
3112:
3109:
3104:
3099:
3096:
3093:
3090:
3085:
3033:
3029:
3025:
3022:
3019:
3014:
3010:
2989:
2963:
2959:
2937:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2911:
2906:
2902:
2898:
2894:
2888:
2884:
2880:
2877:
2872:
2868:
2864:
2860:
2856:
2853:
2850:
2847:
2842:
2837:
2834:
2830:
2809:
2789:
2786:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2736:
2714:
2690:
2681:Given a space
2662:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2626:
2622:
2618:
2614:
2609:
2605:
2600:
2574:
2570:
2566:
2546:
2518:
2490:
2487:
2484:
2479:
2475:
2471:
2468:
2448:
2445:
2442:
2422:
2412:
2398:
2389:and moreover,
2378:
2375:
2370:
2366:
2362:
2359:
2339:
2319:
2299:
2296:
2276:
2252:
2249:
2244:
2240:
2219:
2216:
2213:
2193:
2190:
2170:
2150:
2131:
2128:
2125:
2122:
2102:
2079:
2051:
2029:
2025:
2022:
2016:
2013:
2007:
2001:
1998:
1992:
1986:
1983:
1977:
1974:
1970:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1944:
1940:
1936:
1933:
1930:
1927:
1924:
1921:
1917:
1914:
1911:
1908:
1905:
1902:
1898:
1895:
1892:
1889:
1886:
1883:
1879:
1875:
1870:
1867:
1864:
1861:
1841:
1820:
1815:
1811:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1668:
1647:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1583:
1580:
1577:
1553:
1549:
1546:
1526:
1513:has its usual
1501:
1488:
1485:
1473:
1470:
1448:
1422:
1419:
1399:
1396:
1387:of subsets of
1374:
1352:
1342:
1332:
1329:
1308:
1284:
1280:
1277:
1274:
1271:
1268:
1263:
1257:
1245:cofinal subset
1230:
1208:
1188:
1185:
1182:
1177:
1172:
1167:
1145:
1138:
1132:
1127:
1124:
1116:
1113:
1110:
1104:
1098:
1095:
1092:
1088:
1084:
1081:
1078:
1075:
1070:
1046:
1022:
1000:
978:
967:Equivalently,
954:
951:
931:
911:
891:
888:
885:
882:
860:
855:
852:
832:
829:
826:
823:
818:
813:
810:
790:
787:
784:
779:
774:
769:
743:
734:) for a point
733:
725:
717:
709:
699:
696:
684:
681:
678:
675:
652:
649:
646:
636:
625:
622:
593:
559:
551:
543:
523:
502:
482:
462:
459:
456:
453:
450:
430:
419:if and only if
407:
387:
367:
347:
337:
334:that contains
323:
320:
317:
308:is any subset
307:
296:
276:
249:
246:
237:that contains
226:
206:
183:
163:
149:
134:
131:
119:
116:
105:neighbourhoods
88:
68:
65:
62:
57:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3991:
3980:
3977:
3976:
3974:
3963:
3959:
3955:
3949:
3945:
3941:
3940:Mineola, N.Y.
3937:
3936:
3930:
3926:
3922:
3918:
3912:
3908:
3904:
3900:
3896:
3892:
3888:
3884:
3878:
3874:
3870:
3869:
3864:
3863:
3858:
3854:
3853:
3849:
3839:
3837:9780201087079
3833:
3828:
3827:
3818:
3815:
3811:
3806:
3804:
3800:
3796:
3795:Bourbaki 1989
3791:
3788:
3783:
3781:0-486-66352-3
3777:
3773:
3766:
3763:
3750:
3747:
3731:
3728:
3719:of some point
3718:
3704:
3694:
3690:
3687:
3680:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3632:
3628:
3626:
3612:
3592:
3577:
3574:
3569:
3564:
3560:
3556:
3527:
3524:
3504:
3498:
3495:
3490:
3486:
3465:
3459:
3430:
3427:
3422:
3417:
3413:
3409:
3400:
3360:
3340:
3330:
3316:
3296:
3293:
3284:
3280:
3240:
3237:
3217:
3173:
3170:
3167:
3164:
3161:
3149:
3147:
3145:
3141:
3135:
3122:
3119:
3116:
3110:
3097:
3091:
3073:
3069:
3066:induced by a
3065:
3061:
3057:
3052:
3051:
3047:
3031:
3027:
3023:
3020:
3017:
3012:
3008:
2987:
2979:
2961:
2957:
2935:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2909:
2904:
2900:
2896:
2892:
2886:
2882:
2878:
2875:
2870:
2866:
2862:
2858:
2854:
2848:
2835:
2832:
2828:
2807:
2787:
2784:
2776:
2775:weak topology
2771:
2754:
2748:
2742:
2712:
2704:
2688:
2679:
2677:
2660:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2624:
2620:
2616:
2612:
2607:
2603:
2588:
2572:
2568:
2564:
2544:
2536:
2532:
2516:
2506:
2505:
2501:
2488:
2485:
2482:
2477:
2473:
2469:
2466:
2446:
2443:
2440:
2420:
2410:
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2376:
2373:
2368:
2364:
2360:
2357:
2337:
2317:
2297:
2294:
2274:
2266:
2250:
2247:
2242:
2238:
2217:
2214:
2211:
2191:
2188:
2168:
2148:
2129:
2126:
2123:
2120:
2100:
2093:
2077:
2068:
2066:
2027:
2023:
2020:
2014:
2011:
2005:
1999:
1996:
1990:
1984:
1981:
1975:
1972:
1968:
1958:
1955:
1952:
1949:
1942:
1934:
1928:
1925:
1922:
1915:
1909:
1906:
1903:
1896:
1890:
1887:
1884:
1877:
1868:
1862:
1839:
1813:
1805:
1799:
1796:
1793:
1790:
1783:
1777:
1771:
1765:
1762:
1759:
1756:
1749:
1740:
1737:
1734:
1727:
1721:
1718:
1715:
1712:
1705:
1699:
1696:
1693:
1690:
1645:
1625:
1622:
1619:
1613:
1610:
1607:
1604:
1581:
1578:
1575:
1568:
1547:
1544:
1524:
1516:
1486:
1484:
1471:
1468:
1436:
1435:intersections
1420:
1417:
1397:
1394:
1350:
1336:
1330:
1328:
1326:
1322:
1306:
1299:
1298:partial order
1282:
1278:
1275:
1269:
1255:
1246:
1206:
1183:
1170:
1143:
1136:
1125:
1122:
1114:
1111:
1108:
1102:
1096:
1093:
1090:
1086:
1082:
1076:
998:
965:
952:
949:
929:
909:
889:
886:
883:
880:
853:
850:
830:
824:
811:
808:
785:
772:
757:
741:
727:
719:
711:
703:
697:
695:
682:
676:
666:
665:singleton set
650:
647:
644:
623:
620:
612:
609:
607:
591:
583:
578:
577:
573:
571:
567:
563:
553:
545:
537:
535:
531:
527:
521:
518:
516:
500:
480:
460:
457:
454:
451:
448:
428:
420:
405:
385:
365:
345:
335:
321:
318:
315:
294:
274:
264:neighbourhood
261:
247:
244:
224:
204:
197:
181:
161:
153:
143:
140:
139:
132:
130:
117:
114:
106:
102:
86:
63:
46:
42:
38:
34:
30:
19:
3934:
3902:
3866:
3861:
3850:Bibliography
3825:
3817:
3810:Willard 2004
3790:
3771:
3765:
3749:
3689:
3279:directed set
3153:
3144:pseudometric
3136:
3060:vector space
3058:, that is a
3053:
3049:
3048:
2820:is given by
2772:
2680:
2557:with radius
2531:metric space
2507:
2503:
2502:
2263:denotes the
2069:
2063:denotes the
1490:
1363:is a family
1334:
966:
701:
579:
575:
574:
519:
141:
137:
136:
79:for a point
44:
40:
36:
26:
3072:translation
1567:real number
1327:relation).
756:filter base
714:local basis
608:called the
196:open subset
133:Definitions
33:mathematics
3695:of a point
3681:References
3517:for every
3478:such that
3150:Properties
2978:continuous
2535:open balls
1594:such that
873:such that
730:local base
3859:(1989) .
3590:→
3578:∈
3528:∈
3502:∖
3496:∈
3463:∖
3431:∈
3397:-indexed
3294:⊇
3171:⊆
3165:∈
3062:with the
3021:…
2926:…
2879:ν
2876:−
2863:μ
2836:∈
2833:μ
2808:ν
2701:with the
2657:…
2587:countable
2483:
2470:∈
2444:∈
2374:
2361:∈
2248:
2215:⊆
2124:∈
2024:…
1965:∖
1950:−
1935:∪
1806:∪
1791:−
1772:∪
1757:−
1744:∞
1735:−
1713:−
1691:−
1620:⊆
1605:−
1548:⊆
1307:⊇
1279:⊇
1171:⊆
1156:A family
1126:∈
1112:⊆
1094:⊆
884:⊆
854:∈
812:∈
773:⊆
648:∈
582:non-empty
534:connected
458:⊆
452:∈
319:⊆
3973:Category
3925:10277303
3901:(1984).
3891:18588129
3629:See also
3186:and let
3154:Suppose
3068:seminorm
3064:topology
1487:Examples
1321:superset
584:subset)
29:topology
3871:].
3665:Subbase
3277:into a
2773:In the
2585:form a
2537:around
530:compact
513:in its
194:is any
3962:115240
3960:
3950:
3923:
3913:
3889:
3879:
3834:
3778:
2949:where
2041:where
1325:subset
1106:
1100:
606:filter
526:closed
152:subset
35:, the
3865:[
3309:Then
3253:Make
3054:In a
2529:in a
2310:then
1243:is a
754:is a
604:is a
441:with
99:in a
43:, or
3958:OCLC
3948:ISBN
3921:OCLC
3911:ISBN
3887:OCLC
3877:ISBN
3832:ISBN
3776:ISBN
3625:).
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2897:<
1579:>
718:(or
613:for
336:some
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3399:net
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2181:in
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2067:.
1852::
1658:in
1491:If
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726:or
710:or
522:not
398:in
287:in
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142:An
107:of
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1985:2
1982:1
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