36:
2807:
3479:
and
Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by 1 is
2535:
2903:
The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one another. On the other hand, the underlying topology of a
1788:
3859:
A discrete structure is often used as the "default structure" on a set that doesn't carry any other natural topology, uniformity, or metric; discrete structures can often be used as "extreme" examples to test particular suppositions. For example, any
589:
919:
3880:(or differentiable or analytic manifold) is nothing but a discrete and countable topological space (an uncountable discrete space is not second-countable). We can therefore view any discrete countable group as a 0-dimensional
2026:
3455:
of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps. These facts are examples of a much broader phenomenon, in which discrete structures are usually free on sets.
2094:
3685:
2802:{\displaystyle {\begin{aligned}2^{-(n+1)}&<r\\1&<2^{n+1}r\\r^{-1}&<2^{n+1}\\\log _{2}\left(r^{-1}\right)&<n+1\\-\log _{2}(r)&<n+1\\-1-\log _{2}(r)&<n\end{aligned}}}
1651:
3868:
by giving it the discrete topology, implying that theorems about topological groups apply to all groups. Indeed, analysts may refer to the ordinary, non-topological groups studied by algebraists as "
1921:
1662:
2540:
1407:
490:
2958:
3019:
1224:
1860:
1345:
1186:
3157:
374:
3463:. Certainly the discrete metric space is free when the morphisms are all uniformly continuous maps or all continuous maps, but this says nothing interesting about the metric
834:
3621:
1319:
1070:
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3385:
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1125:
941:
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973:
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4018:
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1482:
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1041:
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1443:
465:
1543:
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2405:
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3821:
3749:
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3705:
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3591:
3567:
3541:
3521:
3501:
3449:
3425:
3316:
3296:
3276:
3107:
3043:
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2849:
2829:
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2150:
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1145:
999:
770:
485:
424:
320:
281:
253:
229:
196:
843:
3954:
1926:
3475:; however, these categories don't have free objects (on more than one element). However, the discrete metric space is free in the category of
2031:
1589:
The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set
3459:
With metric spaces, things are more complicated, because there are several categories of metric spaces, depending on what is chosen for the
3646:
3467:, only the uniform or topological structure. Categories more relevant to the metric structure can be found by limiting the morphisms to
4109:
Pleasants, Peter A.B. (2000). "Designer quasicrystals: Cut-and-project sets with pre-assigned properties". In Baake, Michael (ed.).
1592:
1783:{\textstyle X=\left\{2^{-n}:n\in \mathbb {N} _{0}\right\}=\left\{1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},\dots \right\},}
1865:
4217:
100:
72:
1350:
79:
4179:
4122:
119:
2907:
53:
17:
403:
86:
3400:
57:
4114:
2967:
3025:
and hence not discrete as a uniform space. Nevertheless, it is discrete as a topological space. We say that
4058:
and the space itself). Where the discrete topology is initial or free, the indiscrete topology is final or
1191:
68:
4254:
4166:
4025:
1813:
584:{\displaystyle \rho (x,y)={\begin{cases}1&{\text{if}}\ x\neq y,\\0&{\text{if}}\ x=y\end{cases}}}
3981:
3970:
1324:
1150:
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329:
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924:
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946:
163:
46:
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4161:
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bigger than any given real number, it follows that there will always be at least two points in
2155:
1487:
1447:
779:
723:
687:
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379:
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3199:
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2437:
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93:
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3404:
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1416:
717:
450:
4197:
4132:
8:
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3848:
3768:
3022:
1522:
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914:{\displaystyle S=\left\{{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\ldots \right\}}
4084:
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3873:
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non-discrete uniform or metric space can be discrete; an example is the metric space
979:
773:
141:
4193:
4128:
4089:
4047:
4041:
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2021:{\displaystyle \varepsilon ={\tfrac {1}{2}}\left(x_{n}-x_{n+1}\right)=2^{-(n+2)}.}
4205:
4185:
4171:
4029:
3452:
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808:
induces on it) whose topology is equal to the discrete topology. For example, if
3872:". In some cases, this can be usefully applied, for example in combination with
3896:
3869:
3399:
Any function from a discrete topological space to another topological space is
3192:
3181:
1227:
669:
150:
4238:
4227:
3985:
3403:, and any function from a discrete uniform space to another uniform space is
3392:
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3178:
4079:
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3900:
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1255:
1006:
442:
3428:
3242:
3198:
Combining the above two facts, every discrete uniform or metric space is
3911:
expansion. A product of countably infinite copies of the discrete space
2089:{\displaystyle \left(x_{n}-\varepsilon ,x_{n}+\varepsilon \right)\cap X}
3946:
3912:
3760:
3185:
260:
4055:
3881:
3472:
3356:
3221:
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3680:{\displaystyle \tau ={\mathcal {U}}\cup \left\{\varnothing \right\}}
1790:
consider this set using the usual metric on the real numbers. Then,
35:
3877:
3756:
3460:
3352:
323:
256:
167:
159:
133:
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are open, which is the case if and only if it doesn't contain any
2228:
cannot be uniformly discrete. To see why, suppose there exists an
1656:
Proof that a discrete space is not necessarily uniformly discrete
154:
from each other in a certain sense. The discrete topology is the
4059:
3780:
921:(endowed with the subspace topology) is a discrete subspace of
232:
2132:
Since the intersection of an open set of the real numbers and
3362:
A discrete space is separable if and only if it is countable.
3252:
Every discrete space is metrizable (by the discrete metric).
3021:). This is not the discrete metric; also, this space is not
1646:{\displaystyle \left\{2^{-n}:n\in \mathbb {N} _{0}\right\}.}
4046:
In some ways, the opposite of the discrete topology is the
577:
1916:{\displaystyle (x_{n}-\varepsilon ,x_{n}+\varepsilon ),}
3957:
on the product. Such a homeomorphism is given by using
3166:
Every discrete topological space satisfies each of the
2324:
It suffices to show that there are at least two points
4113:. CRM Monograph Series. Vol. 13. Providence, RI:
3771:
closed but (in contrast to the discrete topology) the
1937:
1665:
889:
874:
859:
162:
in the discrete topology so that in particular, every
4139:
4054:), which has the fewest possible open sets (just the
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3789:
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3095:
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597:
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382:
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308:
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241:
217:
184:
158:
topology that can be given on a set. Every subset is
144:
or similar structure, one in which the points form a
3255:
A finite space is metrizable only if it is discrete.
3071:A topological space is discrete if and only if its
60:. Unsourced material may be challenged and removed.
4012:
3936:
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2152:is open for the induced topology, it follows that
2144:
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2020:
1915:
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1802:
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1511:
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4236:
2851:that are closer to each other than any positive
1402:{\displaystyle x,y\in S,d(x,y)>\varepsilon .}
434:if it is equipped with its discrete uniformity.
4212:. Mineola, New York: Dover Publications, Inc.
4020:is central to the topological approach to the
3298:is a set carrying the discrete topology, then
2953:{\displaystyle X=\{n^{-1}:n\in \mathbb {N} \}}
291:if it is equipped with its discrete topology;
4156:
4007:
3995:
3931:
3919:
3344:(the projection map is the desired covering)
3146:
3116:
2947:
2917:
2172:
2159:
2116:
2103:
1213:
1207:
962:
956:
363:
333:
2407:Since the distance between adjacent points
3973:is necessarily a discrete subspace of its
3191:Every discrete uniform or metric space is
1862:we can surround it with the open interval
1810:is a discrete space, since for each point
4108:
3373:
3170:; in particular, every discrete space is
2943:
1701:
1625:
929:
822:
120:Learn how and when to remove this message
4204:
4145:
4111:Directions in mathematical quasicrystals
4070:an indiscrete space is continuous, etc.
3109:is discrete if and only if the diagonal
3014:{\displaystyle d(x,y)=\left|x-y\right|}
14:
4237:
3907:, with the homeomorphism given by the
3483:Going the other direction, a function
3241:Any two discrete spaces with the same
140:is a particularly simple example of a
2096:is therefore trivially the singleton
1219:{\displaystyle y\in S\setminus \{x\}}
4035:
3854:
3763:, but never both. Said differently,
58:adding citations to reliable sources
29:
3543:is continuous if and only if it is
2384:that are closer to each other than
2185:is open so singletons are open and
24:
4170:(2nd ed.). Berlin, New York:
3658:
3643:can be associated with a topology
3608:
3391:) that is discrete is necessarily
3234:Every non-empty discrete space is
3068:of a discrete space is equal to 0.
1855:{\displaystyle x_{n}=2^{-n}\in X,}
25:
4266:
3851:closed in the discrete topology.
3670:
3549:in the sense that every point in
1340:{\displaystyle \varepsilon >0}
1204:
1181:{\displaystyle d(x,y)>\delta }
3953:to the Cantor set if we use the
3895:copies of the discrete space of
3152:{\displaystyle \{(x,x):x\in X\}}
2960:(with metric inherited from the
2532:that satisfies this inequality:
684:of some given topological space
369:{\displaystyle \{(x,x):x\in X\}}
34:
4028:), which is a weak form of the
3205:Every discrete metric space is
1347:such that for any two distinct
829:{\displaystyle Y:=\mathbb {R} }
45:needs additional citations for
4102:
3616:{\displaystyle {\mathcal {U}}}
3407:. That is, the discrete space
3131:
3119:
2986:
2974:
2782:
2776:
2731:
2725:
2563:
2551:
2494:
2482:
2276:
2264:
2010:
1998:
1907:
1869:
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1555:
1432:
1420:
1387:
1375:
1276:
1264:
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795:
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727:
703:
691:
645:
633:
509:
497:
348:
336:
173:
13:
1:
4115:American Mathematical Society
4095:
2898:
1314:{\displaystyle S\subseteq X,}
1065:{\displaystyle S\subseteq X,}
3796:{\displaystyle \varnothing }
3380:{\displaystyle \mathbb {R} }
3365:Any topological subspace of
3202:if and only if it is finite.
1582:{\displaystyle d(x,y)>r.}
1120:{\displaystyle \delta >0}
936:{\displaystyle \mathbb {R} }
322:is defined by letting every
231:is defined by letting every
7:
4167:Counterexamples in Topology
4073:
4026:Boolean prime ideal theorem
3278:is a topological space and
2894:is not uniformly discrete.
2505:{\displaystyle 2^{-(n+1)},}
2288:{\displaystyle d(x,y)>r}
968:{\displaystyle S\cup \{0\}}
10:
4271:
4039:
3988:properties of products of
3982:foundations of mathematics
3971:locally injective function
3086:for the discrete topology.
2125:{\displaystyle \{x_{n}\}.}
287:discrete topological space
170:in the discrete topology.
3503:from a topological space
3359:is the discrete topology.
3337:{\displaystyle X\times Y}
2811:Since there is always an
2178:{\displaystyle \{x_{n}\}}
1512:{\displaystyle x,y\in E,}
1226:; such a set consists of
801:{\displaystyle (Y,\tau )}
745:{\displaystyle (Y,\tau )}
709:{\displaystyle (Y,\tau )}
651:{\displaystyle (X,\rho )}
619:{\displaystyle x,y\in X.}
395:{\displaystyle X\times X}
27:Type of topological space
3711:non-empty proper subset
3227:Every discrete space is
3212:Every discrete space is
2317:{\displaystyle x\neq y.}
4013:{\displaystyle \{0,1\}}
3945:is homeomorphic to the
3937:{\displaystyle \{0,1\}}
3864:can be considered as a
3707:with the property that
2460:{\displaystyle x_{n+1}}
1094:{\displaystyle x\in S,}
4162:Seebach, J. Arthur Jr.
4014:
3951:uniformly homeomorphic
3938:
3841:
3817:
3797:
3779:open and closed (i.e.
3745:
3725:
3701:
3681:
3637:
3617:
3587:
3563:
3537:
3517:
3497:
3445:
3421:
3381:
3338:
3312:
3292:
3272:
3153:
3103:
3082:The singletons form a
3047:topologically discrete
3039:
3015:
2954:
2888:
2868:
2845:
2825:
2803:
2526:
2506:
2461:
2428:
2401:
2378:
2358:
2338:
2318:
2289:
2248:
2247:{\displaystyle r>0}
2222:
2199:
2179:
2146:
2126:
2090:
2022:
1917:
1856:
1804:
1784:
1647:
1583:
1539:
1513:
1478:
1477:{\displaystyle r>0}
1439:
1403:
1341:
1315:
1286:
1285:{\displaystyle (X,d),}
1244:
1220:
1182:
1141:
1121:
1095:
1066:
1037:
1036:{\displaystyle (X,d),}
995:
969:
937:
915:
830:
802:
766:
746:
710:
652:
620:
585:
481:
461:
430:discrete uniform space
420:
396:
370:
316:
277:
249:
225:
192:
146:discontinuous sequence
4210:Topology for Analysis
4015:
3939:
3842:
3818:
3798:
3746:
3726:
3702:
3682:
3638:
3618:
3588:
3564:
3538:
3518:
3498:
3477:bounded metric spaces
3446:
3422:
3382:
3355:as a subspace of the
3339:
3318:is evenly covered by
3313:
3293:
3273:
3220:if and only if it is
3174:, that is, separated.
3154:
3104:
3066:topological dimension
3040:
3016:
2955:
2889:
2869:
2846:
2826:
2804:
2527:
2507:
2462:
2429:
2427:{\displaystyle x_{n}}
2402:
2379:
2359:
2339:
2319:
2290:
2249:
2223:
2205:is a discrete space.
2200:
2180:
2147:
2127:
2091:
2023:
1918:
1857:
1805:
1785:
1648:
1584:
1540:
1514:
1479:
1440:
1438:{\displaystyle (E,d)}
1404:
1342:
1316:
1287:
1245:
1221:
1183:
1142:
1122:
1096:
1067:
1038:
996:
970:
938:
916:
831:
803:
767:
747:
711:
662:discrete metric space
653:
621:
586:
482:
462:
460:{\displaystyle \rho }
421:
397:
371:
317:
278:
250:
226:
193:
4208:(17 October 2008) .
4066:a topological space
3992:
3916:
3831:
3807:
3787:
3735:
3715:
3691:
3647:
3627:
3603:
3577:
3553:
3527:
3523:to a discrete space
3507:
3487:
3469:Lipschitz continuous
3435:
3411:
3405:uniformly continuous
3369:
3322:
3302:
3282:
3262:
3229:totally disconnected
3177:A discrete space is
3113:
3093:
3029:
2968:
2908:
2878:
2855:
2835:
2815:
2536:
2516:
2471:
2438:
2411:
2388:
2368:
2348:
2328:
2299:
2258:
2232:
2212:
2189:
2156:
2136:
2100:
2032:
1927:
1866:
1814:
1794:
1663:
1593:
1549:
1523:
1488:
1462:
1417:
1351:
1325:
1296:
1261:
1234:
1192:
1151:
1131:
1105:
1076:
1047:
1012:
985:
947:
925:
844:
812:
780:
756:
724:
718:topological subspace
688:
630:
595:
491:
471:
451:
410:
380:
330:
306:
267:
239:
215:
182:
54:improve this article
4117:. pp. 95–141.
4052:indiscrete topology
4024:(equivalently, the
3623:on a non-empty set
3077:accumulation points
3055:metrically discrete
2512:we need to find an
1538:{\displaystyle x=y}
1484:such that, for any
298:discrete uniformity
148:, meaning they are
4255:Topological spaces
4158:Steen, Lynn Arthur
4085:List of topologies
4010:
3955:product uniformity
3934:
3909:continued fraction
3905:irrational numbers
3893:countably infinite
3876:. A 0-dimensional
3874:Pontryagin duality
3837:
3813:
3793:
3741:
3721:
3697:
3677:
3633:
3613:
3583:
3559:
3533:
3513:
3493:
3441:
3417:
3389:Euclidean topology
3377:
3334:
3308:
3288:
3268:
3149:
3099:
3051:uniformly discrete
3035:
3011:
2950:
2884:
2867:{\displaystyle r,}
2864:
2841:
2821:
2799:
2797:
2522:
2502:
2457:
2424:
2400:{\displaystyle r.}
2397:
2374:
2354:
2334:
2314:
2285:
2244:
2218:
2195:
2175:
2142:
2122:
2086:
2018:
1946:
1913:
1852:
1800:
1780:
1657:
1643:
1579:
1535:
1509:
1474:
1451:if there exists a
1448:uniformly discrete
1435:
1399:
1337:
1311:
1282:
1252:uniformly discrete
1240:
1216:
1178:
1137:
1117:
1101:there exists some
1091:
1062:
1033:
991:
965:
933:
911:
898:
883:
868:
838:Euclidean topology
826:
798:
772:together with the
762:
742:
706:
648:
616:
581:
576:
477:
457:
416:
392:
366:
312:
273:
245:
221:
188:
4219:978-0-486-46903-4
4062:: every function
4050:(also called the
4036:Indiscrete spaces
4022:ultrafilter lemma
3961:of numbers. (See
3866:topological group
3855:Examples and uses
3840:{\displaystyle X}
3823:. In comparison,
3816:{\displaystyle X}
3775:subsets that are
3744:{\displaystyle X}
3724:{\displaystyle S}
3700:{\displaystyle X}
3636:{\displaystyle X}
3586:{\displaystyle f}
3562:{\displaystyle Y}
3536:{\displaystyle X}
3516:{\displaystyle Y}
3496:{\displaystyle f}
3444:{\displaystyle X}
3420:{\displaystyle X}
3349:subspace topology
3311:{\displaystyle X}
3291:{\displaystyle Y}
3271:{\displaystyle X}
3216:; it is moreover
3168:separation axioms
3102:{\displaystyle X}
3038:{\displaystyle X}
2887:{\displaystyle X}
2844:{\displaystyle X}
2824:{\displaystyle n}
2525:{\displaystyle n}
2377:{\displaystyle X}
2357:{\displaystyle y}
2337:{\displaystyle x}
2221:{\displaystyle X}
2198:{\displaystyle X}
2145:{\displaystyle X}
2028:The intersection
1945:
1803:{\displaystyle X}
1764:
1751:
1738:
1655:
1243:{\displaystyle S}
1140:{\displaystyle x}
994:{\displaystyle S}
897:
882:
867:
774:subspace topology
765:{\displaystyle Y}
680:discrete subspace
564:
560:
535:
531:
480:{\displaystyle X}
419:{\displaystyle X}
315:{\displaystyle X}
276:{\displaystyle X}
248:{\displaystyle X}
224:{\displaystyle X}
207:discrete topology
191:{\displaystyle X}
142:topological space
130:
129:
122:
104:
16:(Redirected from
4262:
4250:General topology
4231:
4206:Wilansky, Albert
4201:
4149:
4143:
4137:
4136:
4106:
4090:Taxicab geometry
4048:trivial topology
4042:Trivial topology
4019:
4017:
4016:
4011:
3959:ternary notation
3943:
3941:
3940:
3935:
3903:to the space of
3846:
3844:
3843:
3838:
3822:
3820:
3819:
3814:
3802:
3800:
3799:
3794:
3750:
3748:
3747:
3742:
3730:
3728:
3727:
3722:
3706:
3704:
3703:
3698:
3686:
3684:
3683:
3678:
3676:
3662:
3661:
3642:
3640:
3639:
3634:
3622:
3620:
3619:
3614:
3612:
3611:
3592:
3590:
3589:
3584:
3568:
3566:
3565:
3560:
3546:locally constant
3542:
3540:
3539:
3534:
3522:
3520:
3519:
3514:
3502:
3500:
3499:
3494:
3450:
3448:
3447:
3442:
3426:
3424:
3423:
3418:
3387:(with its usual
3386:
3384:
3383:
3378:
3376:
3343:
3341:
3340:
3335:
3317:
3315:
3314:
3309:
3297:
3295:
3294:
3289:
3277:
3275:
3274:
3269:
3218:second-countable
3158:
3156:
3155:
3150:
3108:
3106:
3105:
3100:
3089:A uniform space
3044:
3042:
3041:
3036:
3020:
3018:
3017:
3012:
3010:
3006:
2959:
2957:
2956:
2951:
2946:
2932:
2931:
2893:
2891:
2890:
2885:
2873:
2871:
2870:
2865:
2850:
2848:
2847:
2842:
2830:
2828:
2827:
2822:
2808:
2806:
2805:
2800:
2798:
2772:
2771:
2721:
2720:
2688:
2684:
2683:
2664:
2663:
2650:
2649:
2627:
2626:
2607:
2606:
2567:
2566:
2531:
2529:
2528:
2523:
2511:
2509:
2508:
2503:
2498:
2497:
2466:
2464:
2463:
2458:
2456:
2455:
2433:
2431:
2430:
2425:
2423:
2422:
2406:
2404:
2403:
2398:
2383:
2381:
2380:
2375:
2363:
2361:
2360:
2355:
2343:
2341:
2340:
2335:
2323:
2321:
2320:
2315:
2294:
2292:
2291:
2286:
2253:
2251:
2250:
2245:
2227:
2225:
2224:
2219:
2204:
2202:
2201:
2196:
2184:
2182:
2181:
2176:
2171:
2170:
2151:
2149:
2148:
2143:
2131:
2129:
2128:
2123:
2115:
2114:
2095:
2093:
2092:
2087:
2079:
2075:
2068:
2067:
2049:
2048:
2027:
2025:
2024:
2019:
2014:
2013:
1986:
1982:
1981:
1980:
1962:
1961:
1947:
1938:
1922:
1920:
1919:
1914:
1900:
1899:
1881:
1880:
1861:
1859:
1858:
1853:
1842:
1841:
1826:
1825:
1809:
1807:
1806:
1801:
1789:
1787:
1786:
1781:
1776:
1772:
1765:
1757:
1752:
1744:
1739:
1731:
1715:
1711:
1710:
1709:
1704:
1689:
1688:
1652:
1650:
1649:
1644:
1639:
1635:
1634:
1633:
1628:
1613:
1612:
1588:
1586:
1585:
1580:
1544:
1542:
1541:
1536:
1518:
1516:
1515:
1510:
1483:
1481:
1480:
1475:
1457:
1456:
1444:
1442:
1441:
1436:
1408:
1406:
1405:
1400:
1346:
1344:
1343:
1338:
1321:if there exists
1320:
1318:
1317:
1312:
1291:
1289:
1288:
1283:
1249:
1247:
1246:
1241:
1225:
1223:
1222:
1217:
1187:
1185:
1184:
1179:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1100:
1098:
1097:
1092:
1071:
1069:
1068:
1063:
1042:
1040:
1039:
1034:
1000:
998:
997:
992:
974:
972:
971:
966:
942:
940:
939:
934:
932:
920:
918:
917:
912:
910:
906:
899:
890:
884:
875:
869:
860:
835:
833:
832:
827:
825:
807:
805:
804:
799:
771:
769:
768:
763:
751:
749:
748:
743:
715:
713:
712:
707:
682:
681:
664:
663:
657:
655:
654:
649:
625:
623:
622:
617:
590:
588:
587:
582:
580:
579:
562:
561:
558:
533:
532:
529:
486:
484:
483:
478:
466:
464:
463:
458:
446:
445:
432:
431:
425:
423:
422:
417:
401:
399:
398:
393:
375:
373:
372:
367:
326:of the diagonal
321:
319:
318:
313:
300:
299:
289:
288:
282:
280:
279:
274:
259:(and hence also
254:
252:
251:
246:
230:
228:
227:
222:
209:
208:
197:
195:
194:
189:
164:singleton subset
125:
118:
114:
111:
105:
103:
69:"Discrete space"
62:
38:
30:
21:
4270:
4269:
4265:
4264:
4263:
4261:
4260:
4259:
4235:
4234:
4220:
4182:
4172:Springer-Verlag
4153:
4152:
4144:
4140:
4125:
4107:
4103:
4098:
4076:
4044:
4038:
4030:axiom of choice
3993:
3990:
3989:
3984:, the study of
3917:
3914:
3913:
3897:natural numbers
3870:discrete groups
3857:
3832:
3829:
3828:
3808:
3805:
3804:
3788:
3785:
3784:
3767:subset is open
3736:
3733:
3732:
3716:
3713:
3712:
3692:
3689:
3688:
3666:
3657:
3656:
3648:
3645:
3644:
3628:
3625:
3624:
3607:
3606:
3604:
3601:
3600:
3578:
3575:
3574:
3554:
3551:
3550:
3528:
3525:
3524:
3508:
3505:
3504:
3488:
3485:
3484:
3436:
3433:
3432:
3412:
3409:
3408:
3372:
3370:
3367:
3366:
3323:
3320:
3319:
3303:
3300:
3299:
3283:
3280:
3279:
3263:
3260:
3259:
3236:second category
3214:first-countable
3200:totally bounded
3114:
3111:
3110:
3094:
3091:
3090:
3030:
3027:
3026:
2996:
2992:
2969:
2966:
2965:
2942:
2924:
2920:
2909:
2906:
2905:
2901:
2896:
2879:
2876:
2875:
2856:
2853:
2852:
2836:
2833:
2832:
2816:
2813:
2812:
2796:
2795:
2785:
2767:
2763:
2751:
2750:
2734:
2716:
2712:
2706:
2705:
2689:
2676:
2672:
2668:
2659:
2655:
2652:
2651:
2639:
2635:
2628:
2619:
2615:
2612:
2611:
2596:
2592:
2585:
2579:
2578:
2568:
2547:
2543:
2539:
2537:
2534:
2533:
2517:
2514:
2513:
2478:
2474:
2472:
2469:
2468:
2445:
2441:
2439:
2436:
2435:
2418:
2414:
2412:
2409:
2408:
2389:
2386:
2385:
2369:
2366:
2365:
2349:
2346:
2345:
2329:
2326:
2325:
2300:
2297:
2296:
2259:
2256:
2255:
2233:
2230:
2229:
2213:
2210:
2209:
2190:
2187:
2186:
2166:
2162:
2157:
2154:
2153:
2137:
2134:
2133:
2110:
2106:
2101:
2098:
2097:
2063:
2059:
2044:
2040:
2039:
2035:
2033:
2030:
2029:
1994:
1990:
1970:
1966:
1957:
1953:
1952:
1948:
1936:
1928:
1925:
1924:
1895:
1891:
1876:
1872:
1867:
1864:
1863:
1834:
1830:
1821:
1817:
1815:
1812:
1811:
1795:
1792:
1791:
1756:
1743:
1730:
1723:
1719:
1705:
1700:
1699:
1681:
1677:
1676:
1672:
1664:
1661:
1660:
1629:
1624:
1623:
1605:
1601:
1600:
1596:
1594:
1591:
1590:
1550:
1547:
1546:
1524:
1521:
1520:
1519:one has either
1489:
1486:
1485:
1463:
1460:
1459:
1454:
1453:
1418:
1415:
1414:
1413:A metric space
1411:
1352:
1349:
1348:
1326:
1323:
1322:
1297:
1294:
1293:
1262:
1259:
1258:
1235:
1232:
1231:
1228:isolated points
1193:
1190:
1189:
1152:
1149:
1148:
1132:
1129:
1128:
1106:
1103:
1102:
1077:
1074:
1073:
1048:
1045:
1044:
1013:
1010:
1009:
986:
983:
982:
948:
945:
944:
928:
926:
923:
922:
888:
873:
858:
857:
853:
845:
842:
841:
821:
813:
810:
809:
781:
778:
777:
757:
754:
753:
725:
722:
721:
689:
686:
685:
679:
678:
670:isolated points
661:
660:
631:
628:
627:
596:
593:
592:
575:
574:
557:
555:
549:
548:
528:
526:
516:
515:
492:
489:
488:
472:
469:
468:
452:
449:
448:
440:
439:
429:
428:
411:
408:
407:
381:
378:
377:
331:
328:
327:
307:
304:
303:
297:
296:
286:
285:
268:
265:
264:
240:
237:
236:
216:
213:
212:
206:
205:
183:
180:
179:
176:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
18:Discrete metric
15:
12:
11:
5:
4268:
4258:
4257:
4252:
4247:
4233:
4232:
4218:
4202:
4180:
4151:
4150:
4138:
4123:
4100:
4099:
4097:
4094:
4093:
4092:
4087:
4082:
4075:
4072:
4040:Main article:
4037:
4034:
4009:
4006:
4003:
4000:
3997:
3949:; and in fact
3933:
3930:
3927:
3924:
3921:
3856:
3853:
3836:
3826:
3812:
3792:
3778:
3774:
3766:
3754:
3740:
3720:
3710:
3696:
3675:
3672:
3669:
3665:
3660:
3655:
3652:
3632:
3610:
3582:
3558:
3532:
3512:
3492:
3440:
3416:
3397:
3396:
3375:
3363:
3360:
3345:
3333:
3330:
3327:
3307:
3287:
3267:
3256:
3253:
3250:
3239:
3232:
3225:
3210:
3203:
3196:
3189:
3182:if and only if
3175:
3164:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3098:
3087:
3080:
3069:
3060:Additionally:
3034:
3009:
3005:
3002:
2999:
2995:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2949:
2945:
2941:
2938:
2935:
2930:
2927:
2923:
2919:
2916:
2913:
2900:
2897:
2883:
2863:
2860:
2840:
2820:
2794:
2791:
2788:
2786:
2784:
2781:
2778:
2775:
2770:
2766:
2762:
2759:
2756:
2753:
2752:
2749:
2746:
2743:
2740:
2737:
2735:
2733:
2730:
2727:
2724:
2719:
2715:
2711:
2708:
2707:
2704:
2701:
2698:
2695:
2692:
2690:
2687:
2682:
2679:
2675:
2671:
2667:
2662:
2658:
2654:
2653:
2648:
2645:
2642:
2638:
2634:
2631:
2629:
2625:
2622:
2618:
2614:
2613:
2610:
2605:
2602:
2599:
2595:
2591:
2588:
2586:
2584:
2581:
2580:
2577:
2574:
2571:
2569:
2565:
2562:
2559:
2556:
2553:
2550:
2546:
2542:
2541:
2521:
2501:
2496:
2493:
2490:
2487:
2484:
2481:
2477:
2454:
2451:
2448:
2444:
2421:
2417:
2396:
2393:
2373:
2353:
2333:
2313:
2310:
2307:
2304:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2243:
2240:
2237:
2217:
2194:
2174:
2169:
2165:
2161:
2141:
2121:
2118:
2113:
2109:
2105:
2085:
2082:
2078:
2074:
2071:
2066:
2062:
2058:
2055:
2052:
2047:
2043:
2038:
2017:
2012:
2009:
2006:
2003:
2000:
1997:
1993:
1989:
1985:
1979:
1976:
1973:
1969:
1965:
1960:
1956:
1951:
1944:
1941:
1935:
1932:
1912:
1909:
1906:
1903:
1898:
1894:
1890:
1887:
1884:
1879:
1875:
1871:
1851:
1848:
1845:
1840:
1837:
1833:
1829:
1824:
1820:
1799:
1779:
1775:
1771:
1768:
1763:
1760:
1755:
1750:
1747:
1742:
1737:
1734:
1729:
1726:
1722:
1718:
1714:
1708:
1703:
1698:
1695:
1692:
1687:
1684:
1680:
1675:
1671:
1668:
1654:
1642:
1638:
1632:
1627:
1622:
1619:
1616:
1611:
1608:
1604:
1599:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1534:
1531:
1528:
1508:
1505:
1502:
1499:
1496:
1493:
1473:
1470:
1467:
1455:packing radius
1445:is said to be
1434:
1431:
1428:
1425:
1422:
1410:
1409:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1336:
1333:
1330:
1310:
1307:
1304:
1301:
1281:
1278:
1275:
1272:
1269:
1266:
1239:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1136:
1127:(depending on
1116:
1113:
1110:
1090:
1087:
1084:
1081:
1061:
1058:
1055:
1052:
1032:
1029:
1026:
1023:
1020:
1017:
990:
976:
964:
961:
958:
955:
952:
931:
909:
905:
902:
896:
893:
887:
881:
878:
872:
866:
863:
856:
852:
849:
836:has its usual
824:
820:
817:
797:
794:
791:
788:
785:
761:
741:
738:
735:
732:
729:
705:
702:
699:
696:
693:
674:
647:
644:
641:
638:
635:
615:
612:
609:
606:
603:
600:
578:
573:
570:
567:
556:
554:
551:
550:
547:
544:
541:
538:
527:
525:
522:
521:
519:
514:
511:
508:
505:
502:
499:
496:
487:is defined by
476:
456:
435:
415:
391:
388:
385:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
311:
292:
272:
244:
220:
200:
187:
175:
172:
147:
138:discrete space
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
4267:
4256:
4253:
4251:
4248:
4246:
4243:
4242:
4240:
4229:
4225:
4221:
4215:
4211:
4207:
4203:
4199:
4195:
4191:
4187:
4183:
4181:3-540-90312-7
4177:
4173:
4169:
4168:
4163:
4159:
4155:
4154:
4148:, p. 35.
4147:
4146:Wilansky 2008
4142:
4134:
4130:
4126:
4124:0-8218-2629-8
4120:
4116:
4112:
4105:
4101:
4091:
4088:
4086:
4083:
4081:
4078:
4077:
4071:
4069:
4065:
4061:
4057:
4053:
4049:
4043:
4033:
4031:
4027:
4023:
4004:
4001:
3998:
3987:
3983:
3978:
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3948:
3944:
3928:
3925:
3922:
3910:
3906:
3902:
3898:
3894:
3890:
3885:
3883:
3879:
3875:
3871:
3867:
3863:
3852:
3850:
3834:
3824:
3810:
3790:
3782:
3776:
3772:
3770:
3764:
3762:
3761:closed subset
3758:
3752:
3738:
3718:
3708:
3694:
3673:
3667:
3663:
3653:
3650:
3630:
3599:
3594:
3593:is constant.
3580:
3572:
3556:
3548:
3547:
3530:
3510:
3490:
3481:
3478:
3474:
3470:
3466:
3462:
3457:
3454:
3438:
3430:
3414:
3406:
3402:
3394:
3390:
3364:
3361:
3358:
3354:
3350:
3346:
3331:
3328:
3325:
3305:
3285:
3265:
3257:
3254:
3251:
3248:
3244:
3240:
3237:
3233:
3230:
3226:
3223:
3219:
3215:
3211:
3208:
3204:
3201:
3197:
3194:
3190:
3187:
3183:
3180:
3176:
3173:
3169:
3165:
3162:
3143:
3140:
3137:
3134:
3128:
3125:
3122:
3096:
3088:
3085:
3081:
3078:
3074:
3070:
3067:
3063:
3062:
3061:
3058:
3056:
3052:
3048:
3032:
3024:
3007:
3003:
3000:
2997:
2993:
2989:
2983:
2980:
2977:
2971:
2964:and given by
2963:
2939:
2936:
2933:
2928:
2925:
2921:
2914:
2911:
2895:
2881:
2861:
2858:
2838:
2818:
2809:
2792:
2789:
2787:
2779:
2773:
2768:
2764:
2760:
2757:
2754:
2747:
2744:
2741:
2738:
2736:
2728:
2722:
2717:
2713:
2709:
2702:
2699:
2696:
2693:
2691:
2685:
2680:
2677:
2673:
2669:
2665:
2660:
2656:
2646:
2643:
2640:
2636:
2632:
2630:
2623:
2620:
2616:
2608:
2603:
2600:
2597:
2593:
2589:
2587:
2582:
2575:
2572:
2570:
2560:
2557:
2554:
2548:
2544:
2519:
2499:
2491:
2488:
2485:
2479:
2475:
2452:
2449:
2446:
2442:
2419:
2415:
2394:
2391:
2371:
2351:
2331:
2311:
2308:
2305:
2302:
2282:
2279:
2273:
2270:
2267:
2261:
2241:
2238:
2235:
2215:
2206:
2192:
2167:
2163:
2139:
2119:
2111:
2107:
2083:
2080:
2076:
2072:
2069:
2064:
2060:
2056:
2053:
2050:
2045:
2041:
2036:
2015:
2007:
2004:
2001:
1995:
1991:
1987:
1983:
1977:
1974:
1971:
1967:
1963:
1958:
1954:
1949:
1942:
1939:
1933:
1930:
1910:
1904:
1901:
1896:
1892:
1888:
1885:
1882:
1877:
1873:
1849:
1846:
1843:
1838:
1835:
1831:
1827:
1822:
1818:
1797:
1777:
1773:
1769:
1766:
1761:
1758:
1753:
1748:
1745:
1740:
1735:
1732:
1727:
1724:
1720:
1716:
1712:
1706:
1696:
1693:
1690:
1685:
1682:
1678:
1673:
1669:
1666:
1653:
1640:
1636:
1630:
1620:
1617:
1614:
1609:
1606:
1602:
1597:
1576:
1573:
1570:
1564:
1561:
1558:
1552:
1532:
1529:
1526:
1506:
1503:
1500:
1497:
1494:
1491:
1471:
1468:
1465:
1458:
1450:
1449:
1429:
1426:
1423:
1396:
1393:
1390:
1384:
1381:
1378:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1334:
1331:
1328:
1308:
1305:
1302:
1299:
1279:
1273:
1270:
1267:
1257:
1253:
1237:
1229:
1210:
1201:
1198:
1195:
1175:
1172:
1166:
1163:
1160:
1154:
1134:
1114:
1111:
1108:
1088:
1085:
1082:
1079:
1072:if for every
1059:
1056:
1053:
1050:
1030:
1024:
1021:
1018:
1008:
1004:
988:
981:
977:
959:
953:
950:
907:
903:
900:
894:
891:
885:
879:
876:
870:
864:
861:
854:
850:
847:
839:
818:
815:
792:
789:
786:
775:
759:
752:(a subset of
736:
733:
730:
719:
700:
697:
694:
683:
675:
672:
671:
665:
642:
639:
636:
626:In this case
613:
610:
607:
604:
601:
598:
571:
568:
565:
552:
545:
542:
539:
536:
523:
517:
512:
506:
503:
500:
494:
474:
454:
447:
444:
436:
433:
413:
405:
389:
386:
383:
360:
357:
354:
351:
345:
342:
339:
325:
309:
301:
293:
290:
270:
262:
258:
242:
234:
218:
210:
202:
201:
199:
185:
171:
169:
165:
161:
157:
153:
152:
145:
143:
139:
135:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
4209:
4165:
4141:
4110:
4104:
4080:Cylinder set
4067:
4063:
4051:
4045:
3979:
3963:Cantor space
3901:homeomorphic
3886:
3858:
3595:
3571:neighborhood
3544:
3482:
3458:
3398:
3247:homeomorphic
3059:
3054:
3050:
3046:
2902:
2810:
2207:
1658:
1452:
1446:
1412:
1256:metric space
1251:
1147:) such that
1007:metric space
1002:
716:refers to a
677:
667:
659:
658:is called a
438:
427:
295:
284:
204:
178:Given a set
177:
149:
137:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
3986:compactness
3757:open subset
3598:ultrafilter
3471:maps or to
3431:on the set
3243:cardinality
174:Definitions
4239:Categories
4198:0386.54001
4133:0982.52018
4096:References
3947:Cantor set
3827:subset of
3759:or else a
3473:short maps
3401:continuous
3073:singletons
2899:Properties
2874:therefore
2254:such that
110:March 2011
80:newspapers
4228:227923899
4056:empty set
3965:.) Every
3882:Lie group
3791:∅
3671:∅
3664:∪
3651:τ
3573:on which
3465:structure
3461:morphisms
3393:countable
3357:real line
3329:×
3222:countable
3172:Hausdorff
3161:entourage
3141:∈
3001:−
2962:real line
2940:∈
2926:−
2774:
2761:−
2755:−
2723:
2710:−
2678:−
2666:
2621:−
2549:−
2480:−
2306:≠
2295:whenever
2208:However,
2081:∩
2073:ε
2054:ε
2051:−
1996:−
1964:−
1931:ε
1905:ε
1886:ε
1883:−
1844:∈
1836:−
1770:…
1697:∈
1683:−
1621:∈
1607:−
1501:∈
1394:ε
1364:∈
1329:ε
1303:⊆
1230:. A set
1205:∖
1199:∈
1176:δ
1109:δ
1083:∈
1054:⊆
954:∪
904:…
793:τ
737:τ
701:τ
668:space of
643:ρ
608:∈
540:≠
495:ρ
455:ρ
441:discrete
404:entourage
387:×
358:∈
4245:Topology
4164:(1978).
4074:See also
3878:manifold
3847:is open
3453:category
3353:integers
3193:complete
3049:but not
3023:complete
1188:for all
1003:discrete
591:for any
324:superset
168:open set
151:isolated
134:topology
4190:0507446
3980:In the
3889:product
3480:short.
3451:in the
3351:on the
3207:bounded
3179:compact
1254:in the
975:is not.
263:), and
94:scholar
4226:
4216:
4196:
4188:
4178:
4131:
4121:
4060:cofree
3975:domain
3783:) are
3781:clopen
3753:either
3596:Every
3569:has a
3186:finite
3184:it is
3159:is an
1923:where
563:
534:
443:metric
406:, and
402:be an
261:closed
233:subset
166:is an
156:finest
96:
89:
82:
75:
67:
3969:of a
3967:fiber
3862:group
3825:every
3765:every
3709:every
3084:basis
1005:in a
840:then
776:that
666:or a
426:is a
283:is a
101:JSTOR
87:books
4224:OCLC
4214:ISBN
4176:ISBN
4119:ISBN
4064:from
3803:and
3777:both
3773:only
3429:free
3347:The
3245:are
3064:The
2790:<
2739:<
2694:<
2633:<
2590:<
2573:<
2434:and
2344:and
2280:>
2239:>
1659:Let
1571:>
1469:>
1391:>
1332:>
1292:for
1173:>
1112:>
1043:for
943:but
437:the
294:the
257:open
203:the
160:open
136:, a
73:news
4194:Zbl
4129:Zbl
3977:.
3899:is
3891:of
3849:and
3755:an
3751:is
3731:of
3687:on
3427:is
3258:If
3053:or
3045:is
2765:log
2714:log
2657:log
2467:is
2364:in
1545:or
1250:is
1001:is
980:set
720:of
467:on
376:in
302:on
255:be
235:of
211:on
132:In
56:by
4241::
4222:.
4192:.
4186:MR
4184:.
4174:.
4160:;
4127:.
4068:to
4032:.
3887:A
3884:.
3769:or
3057:.
978:a
819::=
676:a
559:if
530:if
4230:.
4200:.
4135:.
4008:}
4005:1
4002:,
3999:0
3996:{
3932:}
3929:1
3926:,
3923:0
3920:{
3835:X
3811:X
3739:X
3719:S
3695:X
3674:}
3668:{
3659:U
3654:=
3631:X
3609:U
3581:f
3557:Y
3531:X
3511:Y
3491:f
3439:X
3415:X
3395:.
3374:R
3332:Y
3326:X
3306:X
3286:Y
3266:X
3249:.
3238:.
3231:.
3224:.
3209:.
3195:.
3188:.
3163:.
3147:}
3144:X
3138:x
3135::
3132:)
3129:x
3126:,
3123:x
3120:(
3117:{
3097:X
3079:.
3033:X
3008:|
3004:y
2998:x
2994:|
2990:=
2987:)
2984:y
2981:,
2978:x
2975:(
2972:d
2948:}
2944:N
2937:n
2934::
2929:1
2922:n
2918:{
2915:=
2912:X
2882:X
2862:,
2859:r
2839:X
2819:n
2793:n
2783:)
2780:r
2777:(
2769:2
2758:1
2748:1
2745:+
2742:n
2732:)
2729:r
2726:(
2718:2
2703:1
2700:+
2697:n
2686:)
2681:1
2674:r
2670:(
2661:2
2647:1
2644:+
2641:n
2637:2
2624:1
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2609:r
2604:1
2601:+
2598:n
2594:2
2583:1
2576:r
2564:)
2561:1
2558:+
2555:n
2552:(
2545:2
2520:n
2500:,
2495:)
2492:1
2489:+
2486:n
2483:(
2476:2
2453:1
2450:+
2447:n
2443:x
2420:n
2416:x
2395:.
2392:r
2372:X
2352:y
2332:x
2312:.
2309:y
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2268:x
2265:(
2262:d
2242:0
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2193:X
2173:}
2168:n
2164:x
2160:{
2140:X
2120:.
2117:}
2112:n
2108:x
2104:{
2084:X
2077:)
2070:+
2065:n
2061:x
2057:,
2046:n
2042:x
2037:(
2016:.
2011:)
2008:2
2005:+
2002:n
1999:(
1992:2
1988:=
1984:)
1978:1
1975:+
1972:n
1968:x
1959:n
1955:x
1950:(
1943:2
1940:1
1934:=
1911:,
1908:)
1902:+
1897:n
1893:x
1889:,
1878:n
1874:x
1870:(
1850:,
1847:X
1839:n
1832:2
1828:=
1823:n
1819:x
1798:X
1778:,
1774:}
1767:,
1762:8
1759:1
1754:,
1749:4
1746:1
1741:,
1736:2
1733:1
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1725:1
1721:{
1717:=
1713:}
1707:0
1702:N
1694:n
1691::
1686:n
1679:2
1674:{
1670:=
1667:X
1641:.
1637:}
1631:0
1626:N
1618:n
1615::
1610:n
1603:2
1598:{
1577:.
1574:r
1568:)
1565:y
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1559:x
1556:(
1553:d
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1530:=
1527:x
1507:,
1504:E
1498:y
1495:,
1492:x
1472:0
1466:r
1433:)
1430:d
1427:,
1424:E
1421:(
1397:.
1388:)
1385:y
1382:,
1379:x
1376:(
1373:d
1370:,
1367:S
1361:y
1358:,
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1335:0
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1306:X
1300:S
1280:,
1277:)
1274:d
1271:,
1268:X
1265:(
1238:S
1214:}
1211:x
1208:{
1202:S
1196:y
1170:)
1167:y
1164:,
1161:x
1158:(
1155:d
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1115:0
1089:,
1086:S
1080:x
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1057:X
1051:S
1031:,
1028:)
1025:d
1022:,
1019:X
1016:(
989:S
963:}
960:0
957:{
951:S
930:R
908:}
901:,
895:4
892:1
886:,
880:3
877:1
871:,
865:2
862:1
855:{
851:=
848:S
823:R
816:Y
796:)
790:,
787:Y
784:(
760:Y
740:)
734:,
731:Y
728:(
704:)
698:,
695:Y
692:(
673:.
646:)
640:,
637:X
634:(
614:.
611:X
605:y
602:,
599:x
572:y
569:=
566:x
553:0
546:,
543:y
537:x
524:1
518:{
513:=
510:)
507:y
504:,
501:x
498:(
475:X
414:X
390:X
384:X
364:}
361:X
355:x
352::
349:)
346:x
343:,
340:x
337:(
334:{
310:X
271:X
243:X
219:X
198::
186:X
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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