6735:
1859:
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197:, which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A."
4216:
3839:(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be
5143:
we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to
4271:
mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.
3939:
is strictly finer than the
Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.
5003:
or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the
5621:. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them. Examples of such properties include
1402:
The open sets then satisfy the axioms given below in the next definition of a topological space. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining
3014:
There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms.
212:
seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are
2802:
a topology, because (for example) the union of all finite sets not containing zero is not finite and therefore not a member of the family of finite sets. The union of all finite sets not containing zero is also not all of
5371:
3284:
A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms
2475:
1173:
The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of
3381:
2267:
232:
in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by
108:
Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called
6235:
5480:
265:
3875:, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.
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A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given the
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204:'s work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered". "
3835:
in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the
2370:
285:
5696: – a set X equipped with a function that associates to every compact Hausdorff space K a collection of maps K→C satisfying certain natural conditions
70:
formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through
5323:
3301:
are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading.
5950:
4933:
can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any
3751:
of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A
6430:
6131:
5858:. Göschens Lehrbücherei/Gruppe I: Reine und Angewandte Mathematik Serie (in German). Leipzig: Von Veit (published 2011). p. 211.
4732:
is a variant of the
Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every
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228:" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by
5515:
are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
3331:
2189:
6774:
3767:
if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.
182:
3854:. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.
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86:
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237:
205:
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3023:
3019:
999:
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6483:
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6079:
5646:
5622:
5149:
4268:
3851:
3787:
3100:
623:
5649:– The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.
1882:
be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set
6769:
5827:
5284:
4259:
in which the open sets are the intersections of the open sets of the larger space with the subset. For any
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1822:
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295:
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in which the open sets are the empty set and the sets whose complement is finite. This is the smallest
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4012:
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Many topologies can be defined on a set to form a topological space. When every open set of a topology
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109:
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170:
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90:
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has a topology native to it, and this can be extended to vector spaces over that field.
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1406:
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1005:
903:
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793:
705:
685:
633:
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421:
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349:
329:
299:
209:
186:
125:
6811:
6666:
6576:
6556:
6268:
6108:
5675:
5661: – Generalization of the notion of convergence that is found in general topology
5652:
4399:
4235:
3798:
3752:
3748:
3454:
3407:
2007:] is missing; the bottom-right example is not a topology because the intersection of
256:
214:
6759:
6651:
6571:
6517:
5523:
5519:
5244:
4938:
4930:
4917:
4710:
4198:
is the simplest non-discrete topological space. It has important relations to the
174:
129:
98:
5853:
4267:, which is generated by the inverse images of open sets of the factors under the
3794:
are continuous functions. The attempt to classify the objects of this category (
17:
6749:
6661:
6148:
6063:
5164:
5157:
4970:
221:
31:
6605:
6536:
6495:
6218:, (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997).
5541:
5248:
4352:
3847:
2865:
6630:
6231:
3756:
3031:
2600:
1567:
6155:(Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997).
5765:
5366:{\displaystyle \operatorname {cl} \{x\}\subseteq \operatorname {cl} \{y\},}
3293:
are sometimes used in place of finer and coarser, respectively. The terms
1199:
A standard example of such a system of neighbourhoods is for the real line
294:
is shown by the fact that there are several equivalent definitions of this
5877:
Unter einem m e t r i s c h e n R a u m e verstehen wir eine Menge
5570:
consists of the so-called "marked metric graph structures" of volume 1 on
2903:
Using these axioms, another way to define a topological space is as a set
6615:
6583:
6532:
6439:
5522:
has a natural topology that generalizes many of the geometric aspects of
4176:
3862:
3814:
3806:
3791:
3064:
944:
304:
102:
71:
59:
39:
5745:
2470:{\displaystyle \tau =\{\varnothing ,\{2\},\{1,2\},\{2,3\},\{1,2,3\},X\}}
6181:
6100:
6041:
4643:
we construct a basis set consisting of all subsets of the union of the
4184:
3727:
This relates easily to the usual definition in analysis. Equivalently,
302:
suited for the application. The most commonly used is that in terms of
244:
first expressed the idea that a surface is a topological space that is
5131:
have a standard topology in which the basic open sets are open balls.
1551:
6033:
1710:
As this definition of a topology is the most commonly used, the set
4469:
is continuous. A common example of a quotient topology is when an
263:, though it was Hausdorff who popularised the term "metric space" (
5614:
3795:
1857:
627:
236:. His first article on this topic appeared in 1894. In the 1930s,
169:
relating the number of vertices (V), edges (E) and faces (F) of a
67:
2896:
The intersection of any collection of closed sets is also closed.
177:. The study and generalization of this formula, specifically by
6412:
1923:
The bottom-left example is not a topology because the union of
5705: – subset of a topological space whose closure is compact
4209:
3376:{\displaystyle F=\left\{\tau _{\alpha }:\alpha \in A\right\}}
2899:
The union of any finite number of closed sets is also closed.
2863:, the above axioms defining open sets become axioms defining
2262:{\displaystyle \tau =\{\{\},\{1,2,3,4\}\}=\{\varnothing ,X\}}
819:
737:
659:
607:
560:
503:
447:
379:
6408:
4527:
on the set of all non-empty subsets of a topological space
3067:. A topology is completely determined if for every net in
810:
belongs to every one of its neighbourhoods with respect to
46:
is defined but cannot necessarily be measured by a numeric
6237:
3018:
Another way to define a topological space is by using the
3882:. Here, the basic open sets are the half open intervals
5789:
5787:
4223:
It has been suggested that portions of this section be
3304:
The collection of all topologies on a given fixed set
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Every subset of a topological space can be given the
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Pages displaying wikidata descriptions as a fallback
5698:
Pages displaying wikidata descriptions as a fallback
4263:
of topological spaces, the product can be given the
1664:
The intersection of any finite number of members of
6675:
6639:
6525:
6446:
6072:"Moduli of graphs and automorphisms of free groups"
3477:is the meet of the collection of all topologies on
255:in 1914 in his seminal "Principles of Set Theory".
6275:, Prentice Hall; 2nd edition (December 28, 1999).
6012:
5985:
5815:
5714: – Mathematical set with some added structure
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370:, though they can be any mathematical object. Let
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77:A topological space is the most general type of a
4554:, is generated by the following basis: for every
4183:since it is locally Euclidean. Similarly, every
308:, but perhaps more intuitive is that in terms of
74:, which is easier than the others to manipulate.
6022:Proceedings of the American Mathematical Society
5243:the closed sets of the Zariski topology are the
4706:on the set of all non-empty closed subsets of a
5684: – Semicontinuity for set-valued functions
4926:, a precise notion of distance between points.
4516:is then the natural projection onto the set of
5852:(1914) . "Punktmengen in allgemeinen Räumen".
5613:Topological spaces can be broadly classified,
4944:There are many ways of defining a topology on
2986:are the closed sets, and their complements in
58:, along with an additional structure called a
50:. More specifically, a topological space is a
6424:
5747:Introduction to metric and topological spaces
3846:There exist numerous topologies on any given
93:. Common types of topological spaces include
27:Mathematical space with a notion of closeness
8:
5475:{\displaystyle \{\varnothing \}\cup \Gamma }
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4670:that have non-empty intersections with each
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1907:
1889:
6257:, McGraw-Hill; 1st edition (June 1, 1968).
3763:is also continuous. Two spaces are called
2776:of all finite subsets of the integers plus
2289:required by the axioms forms a topology on
1290:if it includes an open interval containing
346:be a (possibly empty) set. The elements of
286:Axiomatic foundations of topological spaces
6792:
6765:
6431:
6417:
6409:
4881:and have nonempty intersections with each
6247:General investigations of curved surfaces
6198:(3rd edition of differently titled books)
6171:Elements of Mathematics: General Topology
6004:
5998:
5977:
5971:
5966:Anderson, B. A.; Stewart, D. G. (1969). "
5928:
5916:
5805:J. Stillwell, Mathematics and its history
5793:
5581:
5575:
5554:
5548:
5487:
5455:
5435:
5411:
5378:
5325:
5299:
5224:
5220:
5219:
5216:
5195:
5191:
5190:
5187:
5115:
5111:
5110:
5107:
5081:
5080:
5078:
5053:
5049:
5048:
5045:
5020:
5016:
5015:
5012:
4999:. The set of all open intervals forms a
4981:
4980:
4978:
4952:
4951:
4949:
4941:this topology is the same for all norms.
4892:
4886:
4866:
4846:
4823:
4803:
4782:
4763:
4757:
4737:
4717:
4681:
4675:
4654:
4648:
4625:
4604:
4585:
4579:
4559:
4532:
4501:
4478:
4454:
4434:
4407:
4383:
4363:
4324:
4304:
4284:
4155:
4135:
4115:
4083:
4048:
4014:
3972:
3948:
3925:
3924:
3922:
3887:
3805:has motivated areas of research, such as
3732:
3694:
3674:
3654:
3625:
3605:
3579:
3541:
3502:
3482:
3462:
3435:
3415:
3388:
3350:
3333:
3309:
3265:
3259:
3234:
3228:
3204:
3198:
3173:
3167:
3143:
3137:
3116:
3110:
3072:
3039:
2991:
2971:
2948:
2928:
2908:
2878:
2833:
2811:
2810:
2808:
2784:
2783:
2781:
2761:
2739:
2738:
2730:
2689:
2666:
2631:
2608:
2584:
2527:
2502:
2482:
2372:
2319:
2294:
2274:
2191:
2168:
2107:
2076:
2044:
2012:
1980:
1954:
1928:
1887:
1867:
1829:
1794:
1764:
1739:
1715:
1689:
1669:
1644:
1624:
1595:
1575:
1535:
1488:
1468:
1448:
1428:
1408:
1384:
1364:
1340:
1320:
1295:
1275:
1252:
1251:
1249:
1229:
1207:
1206:
1204:
1179:
1152:
1132:
1112:
1092:
1072:
1052:
1027:
1007:
978:
952:
925:
905:
882:
862:
842:
818:
817:
815:
795:
766:
736:
735:
727:
707:
687:
658:
657:
655:
635:
606:
605:
603:
582:
559:
558:
556:
536:
502:
501:
499:
476:
446:
445:
443:
423:
403:
378:
377:
375:
351:
331:
251:Topological spaces were first defined by
195:General investigations of curved surfaces
136:
6306:, Macdonald Technical & Scientific,
6290:, Springer; 1st edition (July 6, 2005).
5731:
5678: – Type of topological vector space
5724:
5690: – In mathematics, vector subspace
5460:
5255:Topological spaces with order structure
2383:
2247:
1834:
3063:is a generalisation of the concept of
3022:, which define the closed sets as the
2661:consisting of all possible subsets of
2269:consisting of only the two subsets of
1730:of the open sets is commonly called a
790:In other words, each point of the set
754:{\displaystyle N\in {\mathcal {N}}(x)}
6332:, Holt, Rinehart and Winston (1970).
5904:
5892:
5778:
3568:between topological spaces is called
1562:and satisfying the following axioms:
7:
6255:Schaum's Outline of General Topology
5836:participating institution membership
5603:Classification of topological spaces
4473:is defined on the topological space
3878:The real line can also be given the
1379:is a neighbourhood of all points in
1147:is a neighbourhood of each point of
259:had been defined earlier in 1906 by
5393:denotes an operator satisfying the
5386:{\displaystyle \operatorname {cl} }
4791:{\displaystyle U_{1},\ldots ,U_{n}}
4613:{\displaystyle U_{1},\ldots ,U_{n}}
4191:inherits a natural topology from .
3998:{\displaystyle \gamma =[0,\gamma )}
2684:In this case the topological space
1615:Any arbitrary (finite or infinite)
62:, which can be defined as a set of
6288:A Taste of Topology (Universitext)
6132:Undergraduate Texts in Mathematics
5655: – Type of mathematical space
5469:
5413:
2639:
220:The subject is clearly defined by
81:that allows for the definition of
25:
5672: – Type of topological space
5236:{\displaystyle \mathbb {C} ^{n},}
5135:Topology from algebraic structure
5069:the basic open sets are the open
5036:can be given a topology. In the
4103:{\displaystyle (\alpha ,\gamma )}
4037:{\displaystyle (\alpha ,\beta ),}
3790:are topological spaces and whose
3383:is a collection of topologies on
1315:Given such a structure, a subset
520:{\displaystyle {\mathcal {N}}(x)}
464:{\displaystyle {\mathcal {N}}(x)}
66:for each point that satisfy some
6791:
6764:
6754:
6744:
6733:
6723:
6722:
6516:
5204:{\displaystyle \mathbb {R} ^{n}}
5174:is defined algebraically on the
5124:{\displaystyle \mathbb {C} ^{n}}
5062:{\displaystyle \mathbb {R} ^{n}}
5029:{\displaystyle \mathbb {R} ^{n}}
4378:is the collection of subsets of
4358:, then the quotient topology on
4214:
3720:{\displaystyle f(M)\subseteq N.}
2756:the set of integers, the family
877:and includes a neighbourhood of
290:The utility of the concept of a
5633:. For algebraic invariants see
5269:if and only if it is the prime
3843:where limit points are unique.
2749:{\displaystyle X=\mathbb {Z} ,}
1530:may be defined as a collection
6245:Gauss, Carl Friedrich (1827).
4335:
4227:out into articles titled
4206:Topology from other topologies
4097:
4085:
4062:
4050:
4028:
4016:
3992:
3980:
3901:
3889:
3868:topology on any infinite set.
3821:Examples of topological spaces
3784:category of topological spaces
3705:
3699:
3636:
3630:
3552:
2966:Thus the sets in the topology
2703:
2691:
2648:
2642:
2568:{\displaystyle X=\{1,2,3,4\},}
2360:{\displaystyle X=\{1,2,3,4\},}
2148:{\displaystyle X=\{1,2,3,4\},}
1808:
1796:
947:of a neighbourhood of a point
827:{\displaystyle {\mathcal {N}}}
748:
742:
667:{\displaystyle {\mathcal {N}}}
630:below are satisfied; and then
615:{\displaystyle {\mathcal {N}}}
568:{\displaystyle {\mathcal {N}}}
514:
508:
458:
452:
387:{\displaystyle {\mathcal {N}}}
322:This axiomatization is due to
246:locally like a Euclidean plane
1:
6344:Vaidyanathaswamy, R. (1960).
5945:. Pearson. pp. 317–319.
5402:Topology from other structure
5091:{\displaystyle \mathbb {C} ,}
4962:{\displaystyle \mathbb {R} ,}
3759:that is continuous and whose
3497:that contain every member of
2821:{\displaystyle \mathbb {Z} ,}
2654:{\displaystyle \tau =\wp (X)}
1217:{\displaystyle \mathbb {R} ,}
318:Definition via neighbourhoods
5750:. Oxford : Clarendon Press.
4988:{\displaystyle \mathbb {R} }
4973:. The standard topology on
3932:{\displaystyle \mathbb {R} }
3132:is also open for a topology
2791:{\displaystyle \mathbb {Z} }
1843:{\displaystyle X\setminus C}
1778:{\displaystyle C\subseteq X}
1259:{\displaystyle \mathbb {R} }
973:is again a neighbourhood of
314:and so this is given first.
6397:Encyclopedia of Mathematics
6329:Counterexamples in Topology
5703:Relatively compact subspace
4299:is a topological space and
4071:{\displaystyle [0,\beta ),}
4009:generated by the intervals
2923:together with a collection
6839:
6685:Banach fixed-point theorem
6126:Armstrong, M. A. (1983) .
5744:Sutherland, W. A. (1975).
5606:
4937:. On a finite-dimensional
4915:
4908:is a member of the basis.
4841:the set of all subsets of
4818:and for every compact set
4429:is the finest topology on
4279:is defined as follows: if
3824:
3526:
3277:{\displaystyle \tau _{2}.}
3216:{\displaystyle \tau _{1},}
3155:{\displaystyle \tau _{2},}
3098:
2855:Definition via closed sets
2497:forms another topology of
1916:{\displaystyle \{1,2,3\}.}
283:
238:James Waddell Alexander II
54:whose elements are called
6718:
6514:
6363:Willard, Stephen (2004).
6348:. Chelsea Publishing Co.
5941:Munkres, James R (2015).
5855:GrundzĂĽge der Mengenlehre
5823:Oxford English Dictionary
5395:Kuratowski closure axioms
5150:topological vector spaces
3871:Any set can be given the
3857:Any set can be given the
3852:finite topological spaces
3850:. Such spaces are called
3774:, one of the fundamental
3649:there is a neighbourhood
3243:{\displaystyle \tau _{1}}
3182:{\displaystyle \tau _{2}}
3125:{\displaystyle \tau _{1}}
3020:Kuratowski closure axioms
2709:{\displaystyle (X,\tau )}
1814:{\displaystyle (X,\tau )}
1515:Definition via open sets
1423:to be a neighbourhood of
1087:includes a neighbourhood
1002:of two neighbourhoods of
6302:Schubert, Horst (1968),
6208:, Academic Press (1969).
6173:, Addison-Wesley (1966).
6080:Inventiones Mathematicae
5647:Complete Heyting algebra
5617:homeomorphism, by their
4344:{\displaystyle f:X\to Y}
4166:{\displaystyle \gamma .}
4005:may be endowed with the
3600:and every neighbourhood
3561:{\displaystyle f:X\to Y}
3101:Comparison of topologies
3095:Comparison of topologies
38:is, roughly speaking, a
5828:Oxford University Press
5419:{\displaystyle \Gamma }
5313:{\displaystyle x\leq y}
5286:specialization preorder
5281:Specialization preorder
4922:Metric spaces embody a
4861:that are disjoint from
4123:{\displaystyle \alpha }
3956:{\displaystyle \gamma }
3430:is the intersection of
2828:and so it cannot be in
2064:{\displaystyle \{2,3\}}
2032:{\displaystyle \{1,2\}}
2000:{\displaystyle \{2,3\}}
1505:{\displaystyle x\in U.}
783:{\displaystyle x\in N.}
438:a non-empty collection
298:. Thus one chooses the
230:Johann Benedict Listing
162:{\displaystyle V-E+F=2}
6740:Mathematics portal
6640:Metrics and properties
6626:Second-countable space
6369:. Dover Publications.
6324:Seebach, J. Arthur Jr.
6183:Topology and Groupoids
6014:
5987:
5694:Quasitopological space
5619:topological properties
5594:
5593:{\displaystyle F_{n}.}
5564:
5499:
5476:
5444:
5420:
5387:
5367:
5314:
5237:
5205:
5125:
5092:
5063:
5030:
4989:
4963:
4902:
4875:
4855:
4835:
4812:
4792:
4746:
4726:
4694:
4693:{\displaystyle U_{i}.}
4664:
4637:
4614:
4568:
4544:
4510:
4490:
4463:
4443:
4419:
4392:
4372:
4345:
4313:
4293:
4167:
4144:
4143:{\displaystyle \beta }
4124:
4104:
4072:
4038:
3999:
3957:
3933:
3911:
3910:{\displaystyle [a,b).}
3741:
3721:
3683:
3663:
3643:
3614:
3594:
3593:{\displaystyle x\in X}
3562:
3514:
3491:
3471:
3447:
3424:
3400:
3377:
3318:
3278:
3244:
3217:
3183:
3156:
3126:
3081:
3051:
3000:
2980:
2960:
2937:
2917:
2887:
2845:
2844:{\displaystyle \tau .}
2822:
2792:
2770:
2750:
2710:
2678:
2655:
2620:
2593:
2569:
2514:
2491:
2471:
2361:
2306:
2283:
2263:
2177:
2149:
2098:
2091:
2065:
2033:
2001:
1969:
1943:
1917:
1876:
1854:Examples of topologies
1844:
1815:
1779:
1751:
1724:
1701:
1700:{\displaystyle \tau .}
1678:
1656:
1655:{\displaystyle \tau .}
1633:
1607:
1606:{\displaystyle \tau .}
1584:
1544:
1506:
1477:
1457:
1437:
1417:
1396:
1373:
1349:
1329:
1307:
1284:
1260:
1238:
1218:
1191:
1164:
1141:
1121:
1101:
1081:
1061:
1039:
1022:is a neighbourhood of
1016:
990:
967:
966:{\displaystyle x\in X}
937:
920:is a neighbourhood of
914:
894:
871:
851:
828:
804:
784:
755:
716:
702:is a neighbourhood of
696:
668:
644:
624:neighbourhood topology
616:
591:
569:
545:
521:
488:
465:
432:
412:
388:
360:
340:
296:mathematical structure
273:
189:of topology. In 1827,
163:
6153:Topology and Geometry
6015:
6013:{\displaystyle T_{1}}
5988:
5986:{\displaystyle T_{1}}
5794:Gallier & Xu 2013
5595:
5565:
5563:{\displaystyle F_{n}}
5500:
5477:
5445:
5421:
5388:
5368:
5315:
5238:
5206:
5126:
5093:
5064:
5031:
4990:
4964:
4903:
4901:{\displaystyle U_{i}}
4876:
4856:
4836:
4813:
4793:
4747:
4727:
4695:
4665:
4663:{\displaystyle U_{i}}
4638:
4615:
4569:
4545:
4511:
4491:
4464:
4444:
4420:
4393:
4373:
4346:
4314:
4294:
4200:theory of computation
4168:
4145:
4125:
4105:
4073:
4039:
4000:
3958:
3934:
3912:
3747:is continuous if the
3742:
3722:
3684:
3664:
3644:
3615:
3595:
3563:
3515:
3492:
3472:
3448:
3425:
3401:
3378:
3319:
3279:
3245:
3218:
3184:
3157:
3127:
3082:
3052:
3001:
2981:
2979:{\displaystyle \tau }
2961:
2943:of closed subsets of
2938:
2936:{\displaystyle \tau }
2918:
2888:
2846:
2823:
2793:
2771:
2769:{\displaystyle \tau }
2751:
2711:
2679:
2656:
2621:
2594:
2570:
2515:
2492:
2472:
2362:
2307:
2284:
2264:
2178:
2150:
2092:
2090:{\displaystyle \{2\}}
2066:
2034:
2002:
1970:
1968:{\displaystyle \{3\}}
1944:
1942:{\displaystyle \{2\}}
1918:
1877:
1875:{\displaystyle \tau }
1861:
1845:
1816:
1780:
1752:
1725:
1723:{\displaystyle \tau }
1702:
1679:
1677:{\displaystyle \tau }
1657:
1634:
1632:{\displaystyle \tau }
1608:
1585:
1545:
1543:{\displaystyle \tau }
1507:
1478:
1463:includes an open set
1458:
1438:
1418:
1397:
1374:
1350:
1330:
1308:
1285:
1261:
1239:
1219:
1192:
1165:
1142:
1122:
1102:
1082:
1062:
1040:
1017:
991:
968:
938:
915:
895:
872:
852:
829:
805:
785:
756:
717:
697:
669:
645:
617:
592:
570:
546:
522:
489:
466:
433:
413:
389:
361:
341:
164:
6695:Invariance of domain
6647:Euler characteristic
6621:Bundle (mathematics)
6253:Lipschutz, Seymour;
5997:
5970:
5609:Topological property
5574:
5547:
5486:
5454:
5434:
5410:
5377:
5324:
5298:
5215:
5186:
5106:
5077:
5044:
5011:
4995:is generated by the
4977:
4948:
4885:
4865:
4845:
4822:
4802:
4756:
4736:
4716:
4674:
4647:
4624:
4578:
4558:
4531:
4500:
4477:
4471:equivalence relation
4453:
4433:
4425:In other words, the
4406:
4382:
4362:
4323:
4303:
4283:
4154:
4134:
4114:
4082:
4047:
4013:
3971:
3947:
3921:
3886:
3880:lower limit topology
3873:cocountable topology
3782:, which denotes the
3731:
3693:
3673:
3653:
3642:{\displaystyle f(x)}
3624:
3604:
3578:
3540:
3523:Continuous functions
3501:
3481:
3461:
3434:
3414:
3387:
3332:
3308:
3258:
3227:
3197:
3166:
3136:
3109:
3071:
3038:
2990:
2970:
2947:
2927:
2907:
2877:
2832:
2807:
2780:
2760:
2729:
2688:
2665:
2630:
2626:which is the family
2607:
2583:
2526:
2501:
2481:
2371:
2318:
2293:
2273:
2190:
2167:
2106:
2075:
2043:
2011:
1979:
1953:
1927:
1886:
1866:
1828:
1793:
1763:
1738:
1714:
1688:
1668:
1643:
1623:
1594:
1574:
1534:
1487:
1467:
1447:
1427:
1407:
1383:
1363:
1339:
1319:
1294:
1274:
1248:
1228:
1203:
1178:
1151:
1131:
1111:
1091:
1071:
1051:
1026:
1006:
977:
951:
924:
904:
881:
861:
841:
814:
794:
765:
726:
706:
686:
654:
634:
602:
581:
555:
535:
498:
475:
442:
422:
402:
374:
350:
330:
191:Carl Friedrich Gauss
135:
6705:Tychonoff's theorem
6700:Poincaré conjecture
6454:General (point-set)
6392:"Topological space"
6093:1986InMat..84...91C
5826:(Online ed.).
5712:Space (mathematics)
5513:functional analysis
4935:normed vector space
4518:equivalence classes
3529:Continuous function
3089:accumulation points
3006:are the open sets.
1266:is defined to be a
366:are usually called
6823:Topological spaces
6690:De Rham cohomology
6611:Polyhedral complex
6601:Simplicial complex
6216:Algebraic Topology
6101:10.1007/BF01388734
6010:
5983:
5635:algebraic topology
5590:
5560:
5498:{\displaystyle X.}
5495:
5472:
5440:
5416:
5383:
5363:
5310:
5290:canonical preorder
5283:: In a space the
5271:spectrum of a ring
5233:
5201:
5176:spectrum of a ring
5146:topological groups
5121:
5088:
5059:
5026:
4985:
4959:
4898:
4871:
4851:
4834:{\displaystyle K,}
4831:
4808:
4788:
4742:
4722:
4690:
4660:
4636:{\displaystyle X,}
4633:
4610:
4564:
4543:{\displaystyle X,}
4540:
4506:
4489:{\displaystyle X.}
4486:
4459:
4439:
4418:{\displaystyle f.}
4415:
4388:
4368:
4341:
4309:
4289:
4189:simplicial complex
4163:
4140:
4120:
4100:
4068:
4034:
3995:
3953:
3929:
3907:
3827:List of topologies
3737:
3717:
3679:
3659:
3639:
3610:
3590:
3558:
3513:{\displaystyle F.}
3510:
3487:
3467:
3446:{\displaystyle F,}
3443:
3420:
3399:{\displaystyle X,}
3396:
3373:
3314:
3274:
3240:
3213:
3179:
3152:
3122:
3077:
3050:{\displaystyle X.}
3047:
2996:
2976:
2959:{\displaystyle X.}
2956:
2933:
2913:
2883:
2873:The empty set and
2841:
2818:
2788:
2766:
2746:
2706:
2677:{\displaystyle X.}
2674:
2651:
2619:{\displaystyle X,}
2616:
2589:
2565:
2513:{\displaystyle X.}
2510:
2487:
2477:of six subsets of
2467:
2357:
2305:{\displaystyle X.}
2302:
2279:
2259:
2173:
2145:
2099:
2087:
2061:
2029:
1997:
1965:
1939:
1913:
1872:
1840:
1811:
1775:
1750:{\displaystyle X.}
1747:
1720:
1697:
1674:
1652:
1629:
1603:
1580:
1540:
1502:
1473:
1453:
1433:
1413:
1395:{\displaystyle U.}
1392:
1369:
1345:
1325:
1306:{\displaystyle x.}
1303:
1280:
1256:
1234:
1214:
1190:{\displaystyle X.}
1187:
1163:{\displaystyle M.}
1160:
1137:
1117:
1097:
1077:
1057:
1047:Any neighbourhood
1038:{\displaystyle x.}
1035:
1012:
989:{\displaystyle x.}
986:
963:
936:{\displaystyle x.}
933:
910:
893:{\displaystyle x,}
890:
867:
847:
824:
800:
780:
751:
712:
692:
664:
640:
612:
587:
577:neighbourhoods of
565:
541:
517:
487:{\displaystyle X.}
484:
461:
428:
408:
398:assigning to each
384:
356:
336:
159:
110:point-set topology
79:mathematical space
6805:
6804:
6594:fundamental group
6167:Bourbaki, Nicolas
5952:978-93-325-4953-1
5919:, definition 2.1.
5834:(Subscription or
5659:Convergence space
5631:separation axioms
5482:is a topology on
5443:{\displaystyle X}
5180:algebraic variety
5154:topological rings
5144:concepts such as
5141:algebraic objects
4874:{\displaystyle K}
4854:{\displaystyle X}
4811:{\displaystyle X}
4745:{\displaystyle n}
4725:{\displaystyle X}
4567:{\displaystyle n}
4525:Vietoris topology
4509:{\displaystyle f}
4462:{\displaystyle f}
4442:{\displaystyle Y}
4427:quotient topology
4391:{\displaystyle Y}
4371:{\displaystyle Y}
4319:is a set, and if
4312:{\displaystyle Y}
4292:{\displaystyle X}
4257:subspace topology
4253:
4252:
4248:
4230:Vietoris topology
3917:This topology on
3859:cofinite topology
3833:discrete topology
3740:{\displaystyle f}
3682:{\displaystyle x}
3662:{\displaystyle M}
3613:{\displaystyle N}
3490:{\displaystyle X}
3470:{\displaystyle F}
3423:{\displaystyle F}
3317:{\displaystyle X}
3080:{\displaystyle X}
3010:Other definitions
2999:{\displaystyle X}
2916:{\displaystyle X}
2886:{\displaystyle X}
2592:{\displaystyle X}
2577:discrete topology
2490:{\displaystyle X}
2282:{\displaystyle X}
2176:{\displaystyle X}
1590:itself belong to
1583:{\displaystyle X}
1476:{\displaystyle U}
1456:{\displaystyle N}
1436:{\displaystyle x}
1416:{\displaystyle N}
1372:{\displaystyle U}
1355:is defined to be
1348:{\displaystyle X}
1328:{\displaystyle U}
1283:{\displaystyle x}
1270:of a real number
1237:{\displaystyle N}
1140:{\displaystyle N}
1120:{\displaystyle x}
1100:{\displaystyle M}
1080:{\displaystyle x}
1060:{\displaystyle N}
1015:{\displaystyle x}
913:{\displaystyle N}
870:{\displaystyle X}
850:{\displaystyle N}
803:{\displaystyle X}
715:{\displaystyle x}
695:{\displaystyle N}
676:topological space
643:{\displaystyle X}
590:{\displaystyle x}
544:{\displaystyle x}
431:{\displaystyle X}
411:{\displaystyle x}
359:{\displaystyle X}
339:{\displaystyle X}
187:boosted the study
173:, and hence of a
171:convex polyhedron
40:geometrical space
36:topological space
18:Vietoris topology
16:(Redirected from
6830:
6818:General topology
6795:
6794:
6768:
6767:
6758:
6748:
6738:
6737:
6726:
6725:
6520:
6433:
6426:
6419:
6410:
6405:
6380:
6366:General Topology
6359:
6316:
6250:
6241:
6197:
6145:
6113:
6112:
6076:
6060:
6054:
6053:
6019:
6017:
6016:
6011:
6009:
6008:
5993:-complements of
5992:
5990:
5989:
5984:
5982:
5981:
5963:
5957:
5956:
5938:
5932:
5926:
5920:
5914:
5908:
5902:
5896:
5890:
5884:
5883:
5874:
5872:
5850:Hausdorff, Felix
5846:
5840:
5839:
5831:
5819:
5812:
5806:
5803:
5797:
5791:
5782:
5776:
5770:
5769:
5741:
5735:
5729:
5708:
5699:
5599:
5597:
5596:
5591:
5586:
5585:
5569:
5567:
5566:
5561:
5559:
5558:
5509:linear operators
5504:
5502:
5501:
5496:
5481:
5479:
5478:
5473:
5449:
5447:
5446:
5441:
5425:
5423:
5422:
5417:
5392:
5390:
5389:
5384:
5372:
5370:
5369:
5364:
5319:
5317:
5316:
5311:
5242:
5240:
5239:
5234:
5229:
5228:
5223:
5210:
5208:
5207:
5202:
5200:
5199:
5194:
5172:Zariski topology
5130:
5128:
5127:
5122:
5120:
5119:
5114:
5097:
5095:
5094:
5089:
5084:
5068:
5066:
5065:
5060:
5058:
5057:
5052:
5035:
5033:
5032:
5027:
5025:
5024:
5019:
5006:Euclidean spaces
4994:
4992:
4991:
4986:
4984:
4968:
4966:
4965:
4960:
4955:
4907:
4905:
4904:
4899:
4897:
4896:
4880:
4878:
4877:
4872:
4860:
4858:
4857:
4852:
4840:
4838:
4837:
4832:
4817:
4815:
4814:
4809:
4798:of open sets in
4797:
4795:
4794:
4789:
4787:
4786:
4768:
4767:
4751:
4749:
4748:
4743:
4731:
4729:
4728:
4723:
4699:
4697:
4696:
4691:
4686:
4685:
4669:
4667:
4666:
4661:
4659:
4658:
4642:
4640:
4639:
4634:
4620:of open sets in
4619:
4617:
4616:
4611:
4609:
4608:
4590:
4589:
4573:
4571:
4570:
4565:
4552:Leopold Vietoris
4549:
4547:
4546:
4541:
4515:
4513:
4512:
4507:
4495:
4493:
4492:
4487:
4468:
4466:
4465:
4460:
4448:
4446:
4445:
4440:
4424:
4422:
4421:
4416:
4397:
4395:
4394:
4389:
4377:
4375:
4374:
4369:
4350:
4348:
4347:
4342:
4318:
4316:
4315:
4310:
4298:
4296:
4295:
4290:
4265:product topology
4244:
4218:
4217:
4210:
4196:Sierpiński space
4181:natural topology
4172:
4170:
4169:
4164:
4150:are elements of
4149:
4147:
4146:
4141:
4129:
4127:
4126:
4121:
4109:
4107:
4106:
4101:
4077:
4075:
4074:
4069:
4043:
4041:
4040:
4035:
4004:
4002:
4001:
3996:
3962:
3960:
3959:
3954:
3938:
3936:
3935:
3930:
3928:
3916:
3914:
3913:
3908:
3841:Hausdorff spaces
3837:trivial topology
3746:
3744:
3743:
3738:
3726:
3724:
3723:
3718:
3688:
3686:
3685:
3680:
3668:
3666:
3665:
3660:
3648:
3646:
3645:
3640:
3619:
3617:
3616:
3611:
3599:
3597:
3596:
3591:
3567:
3565:
3564:
3559:
3519:
3517:
3516:
3511:
3496:
3494:
3493:
3488:
3476:
3474:
3473:
3468:
3452:
3450:
3449:
3444:
3429:
3427:
3426:
3421:
3405:
3403:
3402:
3397:
3382:
3380:
3379:
3374:
3372:
3368:
3355:
3354:
3326:complete lattice
3323:
3321:
3320:
3315:
3283:
3281:
3280:
3275:
3270:
3269:
3249:
3247:
3246:
3241:
3239:
3238:
3222:
3220:
3219:
3214:
3209:
3208:
3188:
3186:
3185:
3180:
3178:
3177:
3161:
3159:
3158:
3153:
3148:
3147:
3131:
3129:
3128:
3123:
3121:
3120:
3086:
3084:
3083:
3078:
3056:
3054:
3053:
3048:
3005:
3003:
3002:
2997:
2985:
2983:
2982:
2977:
2965:
2963:
2962:
2957:
2942:
2940:
2939:
2934:
2922:
2920:
2919:
2914:
2892:
2890:
2889:
2884:
2861:de Morgan's laws
2850:
2848:
2847:
2842:
2827:
2825:
2824:
2819:
2814:
2797:
2795:
2794:
2789:
2787:
2775:
2773:
2772:
2767:
2755:
2753:
2752:
2747:
2742:
2715:
2713:
2712:
2707:
2683:
2681:
2680:
2675:
2660:
2658:
2657:
2652:
2625:
2623:
2622:
2617:
2598:
2596:
2595:
2590:
2574:
2572:
2571:
2566:
2519:
2517:
2516:
2511:
2496:
2494:
2493:
2488:
2476:
2474:
2473:
2468:
2366:
2364:
2363:
2358:
2311:
2309:
2308:
2303:
2288:
2286:
2285:
2280:
2268:
2266:
2265:
2260:
2182:
2180:
2179:
2174:
2154:
2152:
2151:
2146:
2096:
2094:
2093:
2088:
2070:
2068:
2067:
2062:
2038:
2036:
2035:
2030:
2006:
2004:
2003:
1998:
1974:
1972:
1971:
1966:
1948:
1946:
1945:
1940:
1922:
1920:
1919:
1914:
1881:
1879:
1878:
1873:
1850:is an open set.
1849:
1847:
1846:
1841:
1820:
1818:
1817:
1812:
1784:
1782:
1781:
1776:
1756:
1754:
1753:
1748:
1729:
1727:
1726:
1721:
1706:
1704:
1703:
1698:
1683:
1681:
1680:
1675:
1661:
1659:
1658:
1653:
1638:
1636:
1635:
1630:
1612:
1610:
1609:
1604:
1589:
1587:
1586:
1581:
1557:
1549:
1547:
1546:
1541:
1529:
1511:
1509:
1508:
1503:
1482:
1480:
1479:
1474:
1462:
1460:
1459:
1454:
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1401:
1399:
1398:
1393:
1378:
1376:
1375:
1370:
1354:
1352:
1351:
1346:
1334:
1332:
1331:
1326:
1312:
1310:
1309:
1304:
1289:
1287:
1286:
1281:
1265:
1263:
1262:
1257:
1255:
1243:
1241:
1240:
1235:
1223:
1221:
1220:
1215:
1210:
1196:
1194:
1193:
1188:
1169:
1167:
1166:
1161:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1066:
1064:
1063:
1058:
1044:
1042:
1041:
1036:
1021:
1019:
1018:
1013:
995:
993:
992:
987:
972:
970:
969:
964:
942:
940:
939:
934:
919:
917:
916:
911:
899:
897:
896:
891:
876:
874:
873:
868:
856:
854:
853:
848:
833:
831:
830:
825:
823:
822:
809:
807:
806:
801:
789:
787:
786:
781:
760:
758:
757:
752:
741:
740:
721:
719:
718:
713:
701:
699:
698:
693:
673:
671:
670:
665:
663:
662:
649:
647:
646:
641:
621:
619:
618:
613:
611:
610:
598:). The function
596:
594:
593:
588:
574:
572:
571:
566:
564:
563:
551:with respect to
550:
548:
547:
542:
526:
524:
523:
518:
507:
506:
494:The elements of
493:
491:
490:
485:
470:
468:
467:
462:
451:
450:
437:
435:
434:
429:
417:
415:
414:
409:
393:
391:
390:
385:
383:
382:
365:
363:
362:
357:
345:
343:
342:
337:
268:
226:Erlangen Program
181:(1789–1857) and
168:
166:
165:
160:
114:general topology
95:Euclidean spaces
21:
6838:
6837:
6833:
6832:
6831:
6829:
6828:
6827:
6808:
6807:
6806:
6801:
6732:
6714:
6710:Urysohn's lemma
6671:
6635:
6521:
6512:
6484:low-dimensional
6442:
6437:
6390:
6387:
6377:
6362:
6356:
6343:
6314:
6301:
6286:Runde, Volker;
6244:
6230:Gallier, Jean;
6229:
6212:Fulton, William
6194:
6176:
6149:Bredon, Glen E.
6142:
6125:
6122:
6117:
6116:
6074:
6068:Vogtmann, Karen
6062:
6061:
6057:
6034:10.2307/2037491
6000:
5995:
5994:
5973:
5968:
5967:
5965:
5964:
5960:
5953:
5940:
5939:
5935:
5927:
5923:
5915:
5911:
5903:
5899:
5891:
5887:
5870:
5868:
5866:
5848:
5847:
5843:
5833:
5814:
5813:
5809:
5804:
5800:
5792:
5785:
5777:
5773:
5758:
5743:
5742:
5738:
5730:
5726:
5721:
5706:
5697:
5688:Linear subspace
5670:Hausdorff space
5643:
5611:
5605:
5577:
5572:
5571:
5550:
5545:
5544:
5484:
5483:
5452:
5451:
5432:
5431:
5408:
5407:
5404:
5375:
5374:
5322:
5321:
5320:if and only if
5296:
5295:
5257:
5218:
5213:
5212:
5189:
5184:
5183:
5137:
5109:
5104:
5103:
5100:complex numbers
5075:
5074:
5047:
5042:
5041:
5014:
5009:
5008:
4975:
4974:
4946:
4945:
4920:
4914:
4888:
4883:
4882:
4863:
4862:
4843:
4842:
4820:
4819:
4800:
4799:
4778:
4759:
4754:
4753:
4734:
4733:
4714:
4713:
4708:locally compact
4677:
4672:
4671:
4650:
4645:
4644:
4622:
4621:
4600:
4581:
4576:
4575:
4556:
4555:
4529:
4528:
4498:
4497:
4475:
4474:
4451:
4450:
4431:
4430:
4404:
4403:
4398:that have open
4380:
4379:
4360:
4359:
4321:
4320:
4301:
4300:
4281:
4280:
4249:
4219:
4215:
4208:
4202:and semantics.
4152:
4151:
4132:
4131:
4112:
4111:
4080:
4079:
4045:
4044:
4011:
4010:
3969:
3968:
3967:, then the set
3945:
3944:
3919:
3918:
3884:
3883:
3866:
3829:
3823:
3811:homology theory
3807:homotopy theory
3772:category theory
3729:
3728:
3691:
3690:
3671:
3670:
3651:
3650:
3622:
3621:
3602:
3601:
3576:
3575:
3538:
3537:
3531:
3525:
3499:
3498:
3479:
3478:
3459:
3458:
3432:
3431:
3412:
3411:
3385:
3384:
3346:
3345:
3341:
3330:
3329:
3306:
3305:
3261:
3256:
3255:
3230:
3225:
3224:
3200:
3195:
3194:
3169:
3164:
3163:
3139:
3134:
3133:
3112:
3107:
3106:
3103:
3097:
3087:the set of its
3069:
3068:
3036:
3035:
3012:
2988:
2987:
2968:
2967:
2945:
2944:
2925:
2924:
2905:
2904:
2875:
2874:
2857:
2830:
2829:
2805:
2804:
2778:
2777:
2758:
2757:
2727:
2726:
2686:
2685:
2663:
2662:
2628:
2627:
2605:
2604:
2581:
2580:
2524:
2523:
2499:
2498:
2479:
2478:
2369:
2368:
2316:
2315:
2291:
2290:
2271:
2270:
2188:
2187:
2165:
2164:
2104:
2103:
2073:
2072:
2041:
2040:
2009:
2008:
1977:
1976:
1951:
1950:
1925:
1924:
1884:
1883:
1864:
1863:
1856:
1826:
1825:
1791:
1790:
1761:
1760:
1736:
1735:
1712:
1711:
1686:
1685:
1666:
1665:
1641:
1640:
1621:
1620:
1592:
1591:
1572:
1571:
1555:
1532:
1531:
1527:
1517:
1485:
1484:
1465:
1464:
1445:
1444:
1425:
1424:
1405:
1404:
1381:
1380:
1361:
1360:
1337:
1336:
1317:
1316:
1292:
1291:
1272:
1271:
1246:
1245:
1226:
1225:
1224:where a subset
1201:
1200:
1176:
1175:
1149:
1148:
1129:
1128:
1109:
1108:
1089:
1088:
1069:
1068:
1049:
1048:
1024:
1023:
1004:
1003:
975:
974:
949:
948:
922:
921:
902:
901:
879:
878:
859:
858:
857:is a subset of
839:
838:
812:
811:
792:
791:
763:
762:
724:
723:
704:
703:
684:
683:
652:
651:
632:
631:
600:
599:
579:
578:
553:
552:
533:
532:
527:will be called
496:
495:
473:
472:
440:
439:
420:
419:
400:
399:
372:
371:
348:
347:
328:
327:
324:Felix Hausdorff
320:
288:
282:
274:metrischer Raum
264:
261:Maurice Fréchet
253:Felix Hausdorff
242:Hassler Whitney
133:
132:
128:discovered the
122:
28:
23:
22:
15:
12:
11:
5:
6836:
6834:
6826:
6825:
6820:
6810:
6809:
6803:
6802:
6800:
6799:
6789:
6788:
6787:
6782:
6777:
6762:
6752:
6742:
6730:
6719:
6716:
6715:
6713:
6712:
6707:
6702:
6697:
6692:
6687:
6681:
6679:
6673:
6672:
6670:
6669:
6664:
6659:
6657:Winding number
6654:
6649:
6643:
6641:
6637:
6636:
6634:
6633:
6628:
6623:
6618:
6613:
6608:
6603:
6598:
6597:
6596:
6591:
6589:homotopy group
6581:
6580:
6579:
6574:
6569:
6564:
6559:
6549:
6544:
6539:
6529:
6527:
6523:
6522:
6515:
6513:
6511:
6510:
6505:
6500:
6499:
6498:
6488:
6487:
6486:
6476:
6471:
6466:
6461:
6456:
6450:
6448:
6444:
6443:
6438:
6436:
6435:
6428:
6421:
6413:
6407:
6406:
6386:
6385:External links
6383:
6382:
6381:
6375:
6360:
6354:
6341:
6320:Steen, Lynn A.
6317:
6312:
6299:
6284:
6269:Munkres, James
6266:
6251:
6242:
6227:
6209:
6199:
6192:
6174:
6164:
6146:
6140:
6128:Basic Topology
6121:
6118:
6115:
6114:
6055:
6007:
6003:
5980:
5976:
5958:
5951:
5933:
5931:, theorem 2.6.
5929:Armstrong 1983
5921:
5917:Armstrong 1983
5909:
5907:, section 2.2.
5897:
5895:, section 2.1.
5885:
5864:
5841:
5817:"metric space"
5807:
5798:
5783:
5771:
5756:
5736:
5723:
5722:
5720:
5717:
5716:
5715:
5709:
5700:
5691:
5685:
5682:Hemicontinuity
5679:
5673:
5667:
5665:Exterior space
5662:
5656:
5650:
5642:
5639:
5629:, and various
5607:Main article:
5604:
5601:
5589:
5584:
5580:
5557:
5553:
5494:
5491:
5471:
5468:
5465:
5462:
5459:
5439:
5415:
5403:
5400:
5399:
5398:
5382:
5362:
5359:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5309:
5306:
5303:
5294:is defined by
5278:
5256:
5253:
5247:of systems of
5232:
5227:
5222:
5198:
5193:
5136:
5133:
5118:
5113:
5087:
5083:
5073:. Similarly,
5056:
5051:
5038:usual topology
5023:
5018:
4997:open intervals
4983:
4958:
4954:
4916:Main article:
4913:
4910:
4895:
4891:
4870:
4850:
4830:
4827:
4807:
4785:
4781:
4777:
4774:
4771:
4766:
4762:
4741:
4721:
4689:
4684:
4680:
4657:
4653:
4632:
4629:
4607:
4603:
4599:
4596:
4593:
4588:
4584:
4563:
4539:
4536:
4505:
4485:
4482:
4458:
4438:
4414:
4411:
4400:inverse images
4387:
4367:
4340:
4337:
4334:
4331:
4328:
4308:
4288:
4277:quotient space
4261:indexed family
4251:
4250:
4222:
4220:
4213:
4207:
4204:
4162:
4159:
4139:
4119:
4099:
4096:
4093:
4090:
4087:
4067:
4064:
4061:
4058:
4055:
4052:
4033:
4030:
4027:
4024:
4021:
4018:
4007:order topology
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3965:ordinal number
3952:
3927:
3906:
3903:
3900:
3897:
3894:
3891:
3864:
3822:
3819:
3766:
3736:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3678:
3658:
3638:
3635:
3632:
3629:
3609:
3589:
3586:
3583:
3557:
3554:
3551:
3548:
3545:
3527:Main article:
3524:
3521:
3509:
3506:
3486:
3466:
3442:
3439:
3419:
3395:
3392:
3371:
3367:
3364:
3361:
3358:
3353:
3349:
3344:
3340:
3337:
3313:
3300:
3296:
3292:
3288:
3273:
3268:
3264:
3253:
3237:
3233:
3212:
3207:
3203:
3192:
3176:
3172:
3162:one says that
3151:
3146:
3142:
3119:
3115:
3099:Main article:
3096:
3093:
3091:is specified.
3076:
3046:
3043:
3011:
3008:
2995:
2975:
2955:
2952:
2932:
2912:
2901:
2900:
2897:
2894:
2882:
2856:
2853:
2852:
2851:
2840:
2837:
2817:
2813:
2801:
2786:
2765:
2745:
2741:
2737:
2734:
2723:
2719:discrete space
2705:
2702:
2699:
2696:
2693:
2673:
2670:
2650:
2647:
2644:
2641:
2638:
2635:
2615:
2612:
2588:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2520:
2509:
2506:
2486:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2312:
2301:
2298:
2278:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2172:
2162:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2097:], is missing.
2086:
2083:
2080:
2060:
2057:
2054:
2051:
2048:
2028:
2025:
2022:
2019:
2016:
1996:
1993:
1990:
1987:
1984:
1964:
1961:
1958:
1938:
1935:
1932:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1871:
1855:
1852:
1839:
1836:
1833:
1810:
1807:
1804:
1801:
1798:
1788:
1785:is said to be
1774:
1771:
1768:
1746:
1743:
1719:
1708:
1707:
1696:
1693:
1673:
1662:
1651:
1648:
1628:
1619:of members of
1613:
1602:
1599:
1579:
1539:
1516:
1513:
1501:
1498:
1495:
1492:
1472:
1452:
1432:
1412:
1391:
1388:
1368:
1344:
1324:
1302:
1299:
1279:
1269:
1254:
1233:
1213:
1209:
1186:
1183:
1171:
1170:
1159:
1156:
1136:
1116:
1096:
1076:
1056:
1045:
1034:
1031:
1011:
996:
985:
982:
962:
959:
956:
932:
929:
909:
889:
886:
866:
846:
835:
821:
799:
779:
776:
773:
770:
750:
747:
744:
739:
734:
731:
711:
691:
661:
639:
609:
597:
586:
562:
540:
530:
529:neighbourhoods
516:
513:
510:
505:
483:
480:
471:of subsets of
460:
457:
454:
449:
427:
407:
381:
369:
355:
335:
319:
316:
313:
311:neighbourhoods
307:
300:axiomatization
284:Main article:
281:
278:
234:Henri Poincaré
158:
155:
152:
149:
146:
143:
140:
126:Leonhard Euler
121:
118:
64:neighbourhoods
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6835:
6824:
6821:
6819:
6816:
6815:
6813:
6798:
6790:
6786:
6783:
6781:
6778:
6776:
6773:
6772:
6771:
6763:
6761:
6757:
6753:
6751:
6747:
6743:
6741:
6736:
6731:
6729:
6721:
6720:
6717:
6711:
6708:
6706:
6703:
6701:
6698:
6696:
6693:
6691:
6688:
6686:
6683:
6682:
6680:
6678:
6674:
6668:
6667:Orientability
6665:
6663:
6660:
6658:
6655:
6653:
6650:
6648:
6645:
6644:
6642:
6638:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6599:
6595:
6592:
6590:
6587:
6586:
6585:
6582:
6578:
6575:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6555:
6554:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6534:
6531:
6530:
6528:
6524:
6519:
6509:
6506:
6504:
6503:Set-theoretic
6501:
6497:
6494:
6493:
6492:
6489:
6485:
6482:
6481:
6480:
6477:
6475:
6472:
6470:
6467:
6465:
6464:Combinatorial
6462:
6460:
6457:
6455:
6452:
6451:
6449:
6445:
6441:
6434:
6429:
6427:
6422:
6420:
6415:
6414:
6411:
6403:
6399:
6398:
6393:
6389:
6388:
6384:
6378:
6376:0-486-43479-6
6372:
6368:
6367:
6361:
6357:
6351:
6347:
6342:
6339:
6338:0-03-079485-4
6335:
6331:
6330:
6325:
6321:
6318:
6315:
6313:0-356-02077-0
6309:
6305:
6300:
6297:
6296:0-387-25790-X
6293:
6289:
6285:
6282:
6281:0-13-181629-2
6278:
6274:
6270:
6267:
6264:
6263:0-07-037988-2
6260:
6256:
6252:
6248:
6243:
6239:
6238:
6233:
6228:
6225:
6224:0-387-94327-7
6221:
6217:
6213:
6210:
6207:
6203:
6200:
6195:
6193:1-4196-2722-8
6189:
6186:. Booksurge.
6185:
6184:
6179:
6178:Brown, Ronald
6175:
6172:
6168:
6165:
6162:
6161:0-387-97926-3
6158:
6154:
6150:
6147:
6143:
6141:0-387-90839-0
6137:
6133:
6129:
6124:
6123:
6119:
6110:
6106:
6102:
6098:
6094:
6090:
6087:(1): 91–119.
6086:
6082:
6081:
6073:
6069:
6065:
6059:
6056:
6051:
6047:
6043:
6039:
6035:
6031:
6027:
6023:
6020:topologies".
6005:
6001:
5978:
5974:
5962:
5959:
5954:
5948:
5944:
5937:
5934:
5930:
5925:
5922:
5918:
5913:
5910:
5906:
5901:
5898:
5894:
5889:
5886:
5882:
5880:
5867:
5865:9783110989854
5861:
5857:
5856:
5851:
5845:
5842:
5837:
5829:
5825:
5824:
5818:
5811:
5808:
5802:
5799:
5795:
5790:
5788:
5784:
5780:
5775:
5772:
5767:
5763:
5759:
5757:0-19-853155-9
5753:
5749:
5748:
5740:
5737:
5733:
5732:Schubert 1968
5728:
5725:
5718:
5713:
5710:
5704:
5701:
5695:
5692:
5689:
5686:
5683:
5680:
5677:
5676:Hilbert space
5674:
5671:
5668:
5666:
5663:
5660:
5657:
5654:
5653:Compact space
5651:
5648:
5645:
5644:
5640:
5638:
5636:
5632:
5628:
5624:
5623:connectedness
5620:
5616:
5610:
5602:
5600:
5587:
5582:
5578:
5555:
5551:
5543:
5539:
5535:
5533:
5529:
5525:
5521:
5516:
5514:
5510:
5507:Many sets of
5505:
5492:
5489:
5466:
5437:
5429:
5401:
5396:
5380:
5360:
5354:
5348:
5345:
5342:
5336:
5330:
5327:
5307:
5304:
5301:
5293:
5291:
5287:
5282:
5279:
5276:
5272:
5268:
5267:
5263:: A space is
5262:
5259:
5258:
5254:
5252:
5250:
5246:
5245:solution sets
5230:
5225:
5196:
5181:
5177:
5173:
5168:
5166:
5161:
5159:
5155:
5151:
5147:
5142:
5134:
5132:
5116:
5101:
5085:
5072:
5054:
5039:
5021:
5007:
5002:
4998:
4972:
4956:
4942:
4940:
4936:
4932:
4927:
4925:
4919:
4912:Metric spaces
4911:
4909:
4893:
4889:
4868:
4848:
4828:
4825:
4805:
4783:
4779:
4775:
4772:
4769:
4764:
4760:
4739:
4719:
4712:
4709:
4705:
4704:Fell topology
4700:
4687:
4682:
4678:
4655:
4651:
4630:
4627:
4605:
4601:
4597:
4594:
4591:
4586:
4582:
4561:
4553:
4537:
4534:
4526:
4521:
4519:
4503:
4483:
4480:
4472:
4456:
4436:
4428:
4412:
4409:
4401:
4385:
4365:
4357:
4354:
4338:
4332:
4329:
4326:
4306:
4286:
4278:
4273:
4270:
4266:
4262:
4258:
4247:
4242:
4238:
4237:
4236:Fell topology
4232:
4231:
4226:
4221:
4212:
4211:
4205:
4203:
4201:
4197:
4192:
4190:
4186:
4182:
4178:
4173:
4160:
4157:
4137:
4117:
4094:
4091:
4088:
4065:
4059:
4056:
4053:
4031:
4025:
4022:
4019:
4008:
3989:
3986:
3983:
3977:
3974:
3966:
3950:
3941:
3904:
3898:
3895:
3892:
3881:
3876:
3874:
3869:
3867:
3860:
3855:
3853:
3849:
3844:
3842:
3838:
3834:
3828:
3820:
3818:
3816:
3812:
3808:
3804:
3800:
3799:homeomorphism
3797:
3793:
3789:
3785:
3781:
3777:
3773:
3768:
3764:
3762:
3758:
3754:
3753:homeomorphism
3750:
3749:inverse image
3734:
3714:
3711:
3708:
3702:
3696:
3676:
3656:
3633:
3627:
3607:
3587:
3584:
3581:
3574:if for every
3573:
3572:
3555:
3549:
3546:
3543:
3536:
3530:
3522:
3520:
3507:
3504:
3484:
3464:
3456:
3440:
3437:
3417:
3409:
3393:
3390:
3369:
3365:
3362:
3359:
3356:
3351:
3347:
3342:
3338:
3335:
3327:
3311:
3302:
3298:
3294:
3290:
3286:
3271:
3266:
3262:
3251:
3235:
3231:
3210:
3205:
3201:
3190:
3174:
3170:
3149:
3144:
3140:
3117:
3113:
3102:
3094:
3092:
3090:
3074:
3066:
3062:
3057:
3044:
3041:
3033:
3029:
3025:
3021:
3016:
3009:
3007:
2993:
2973:
2953:
2950:
2930:
2910:
2898:
2895:
2880:
2872:
2871:
2870:
2868:
2867:
2862:
2854:
2838:
2835:
2815:
2799:
2763:
2743:
2735:
2732:
2724:
2721:
2720:
2700:
2697:
2694:
2671:
2668:
2645:
2636:
2633:
2613:
2610:
2602:
2586:
2578:
2562:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2532:
2529:
2521:
2507:
2504:
2484:
2461:
2458:
2452:
2449:
2446:
2443:
2440:
2434:
2428:
2425:
2422:
2416:
2410:
2407:
2404:
2398:
2392:
2386:
2377:
2374:
2354:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2324:
2321:
2313:
2299:
2296:
2276:
2253:
2250:
2241:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2208:
2196:
2193:
2186:
2170:
2160:
2158:
2142:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2112:
2109:
2101:
2100:
2081:
2055:
2052:
2049:
2023:
2020:
2017:
1991:
1988:
1985:
1959:
1933:
1910:
1904:
1901:
1898:
1895:
1892:
1869:
1860:
1853:
1851:
1837:
1831:
1824:
1805:
1802:
1799:
1786:
1772:
1769:
1766:
1757:
1744:
1741:
1733:
1717:
1694:
1691:
1671:
1663:
1649:
1646:
1626:
1618:
1614:
1600:
1597:
1577:
1569:
1565:
1564:
1563:
1561:
1553:
1537:
1526:
1522:
1514:
1512:
1499:
1496:
1493:
1490:
1470:
1450:
1430:
1410:
1389:
1386:
1366:
1358:
1342:
1322:
1313:
1300:
1297:
1277:
1268:neighbourhood
1267:
1231:
1211:
1197:
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1157:
1154:
1134:
1114:
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1054:
1046:
1032:
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1009:
1001:
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930:
927:
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884:
864:
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771:
768:
745:
732:
729:
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681:
680:
679:
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637:
629:
625:
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576:
575:(or, simply,
538:
528:
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367:
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325:
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315:
312:
309:
306:
303:
301:
297:
293:
287:
279:
277:
275:
271:
267:
262:
258:
257:Metric spaces
254:
249:
247:
243:
239:
235:
231:
227:
223:
218:
216:
211:
207:
203:
198:
196:
192:
188:
185:(1750–1840),
184:
180:
176:
172:
156:
153:
150:
147:
144:
141:
138:
131:
127:
124:Around 1735,
119:
117:
115:
111:
106:
104:
100:
99:metric spaces
96:
92:
91:connectedness
88:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
6797:Publications
6662:Chern number
6652:Betti number
6551:
6535: /
6526:Key concepts
6474:Differential
6395:
6365:
6346:Set Topology
6345:
6327:
6303:
6287:
6272:
6254:
6246:
6236:
6215:
6205:
6202:ÄŚech, Eduard
6182:
6170:
6152:
6134:. Springer.
6127:
6120:Bibliography
6084:
6078:
6064:Culler, Marc
6058:
6025:
6021:
5961:
5942:
5936:
5924:
5912:
5900:
5888:
5878:
5876:
5869:. Retrieved
5854:
5844:
5821:
5810:
5801:
5774:
5746:
5739:
5727:
5612:
5536:
5520:linear graph
5517:
5506:
5405:
5289:
5285:
5280:
5264:
5260:
5169:
5162:
5158:local fields
5138:
5037:
4971:real numbers
4943:
4939:vector space
4931:metric space
4928:
4921:
4918:Metric space
4711:Polish space
4703:
4701:
4524:
4522:
4274:
4254:
4245:
4234:
4228:
4193:
4174:
3942:
3877:
3870:
3856:
3845:
3830:
3779:
3769:
3765:homeomorphic
3569:
3532:
3303:
3104:
3058:
3024:fixed points
3017:
3013:
2902:
2864:
2858:
2717:
2716:is called a
2163:topology on
1758:
1731:
1709:
1559:
1520:
1518:
1356:
1314:
1198:
1172:
1000:intersection
943:I.e., every
675:
674:is called a
622:is called a
321:
291:
289:
250:
219:
215:homeomorphic
200:Yet, "until
199:
194:
175:planar graph
123:
107:
76:
35:
29:
6760:Wikiversity
6677:Key results
6240:. Springer.
5627:compactness
5538:Outer space
5251:equations.
5165:local field
5098:the set of
4969:the set of
4246:(June 2024)
2893:are closed.
2866:closed sets
2367:the family
1684:belongs to
1639:belongs to
418:(point) in
280:Definitions
222:Felix Klein
32:mathematics
6812:Categories
6606:CW complex
6547:Continuity
6537:Closed set
6496:cohomology
6355:0486404560
6232:Xu, Dianna
6206:Point Sets
5905:Brown 2006
5893:Brown 2006
5838:required.)
5779:Gauss 1827
5542:free group
5249:polynomial
4550:named for
4449:for which
4353:surjective
4269:projection
4187:and every
3848:finite set
3825:See also:
3803:invariants
3776:categories
3689:such that
3571:continuous
2798:itself is
2161:indiscrete
1823:complement
1483:such that
1127:such that
193:published
87:continuity
6785:geometric
6780:algebraic
6631:Cobordism
6567:Hausdorff
6562:connected
6479:Geometric
6469:Continuum
6459:Algebraic
6402:EMS Press
6109:122869546
6028:: 77–81.
5871:20 August
5719:Citations
5470:Γ
5467:∪
5461:∅
5430:on a set
5414:Γ
5349:
5343:⊆
5331:
5305:≤
5277:theorem).
4773:…
4595:…
4336:→
4158:γ
4138:β
4118:α
4095:γ
4089:α
4060:β
4026:β
4020:α
3990:γ
3975:γ
3951:γ
3792:morphisms
3757:bijection
3709:⊆
3585:∈
3553:→
3406:then the
3363:∈
3360:α
3352:α
3348:τ
3263:τ
3232:τ
3202:τ
3171:τ
3141:τ
3114:τ
3032:power set
2974:τ
2931:τ
2836:τ
2764:τ
2701:τ
2640:℘
2634:τ
2601:power set
2384:∅
2375:τ
2248:∅
2194:τ
1870:τ
1835:∖
1806:τ
1770:⊆
1759:A subset
1718:τ
1692:τ
1672:τ
1647:τ
1627:τ
1598:τ
1568:empty set
1560:open sets
1558:, called
1538:τ
1494:∈
958:∈
772:∈
733:∈
305:open sets
266:‹See Tfd›
217:or not."
183:L'Huilier
142:−
103:manifolds
72:open sets
44:closeness
42:in which
6750:Wikibook
6728:Category
6616:Manifold
6584:Homotopy
6542:Interior
6533:Open set
6491:Homology
6440:Topology
6304:Topology
6273:Topology
6234:(2013).
6180:(2006).
6070:(1986).
5943:Topology
5641:See also
5528:vertices
5275:Hochster
5266:spectral
5261:Spectral
5139:For any
4496:The map
4356:function
4177:manifold
3815:K-theory
3535:function
3453:and the
3324:forms a
3295:stronger
3065:sequence
3028:operator
1732:topology
1521:topology
945:superset
761:), then
396:function
292:topology
224:in his "
60:topology
48:distance
6775:general
6577:uniform
6557:compact
6508:Digital
6404:, 2001
6089:Bibcode
6050:0244927
6042:2037491
5766:1679102
5734:, p. 13
4752:-tuple
4574:-tuple
4241:Discuss
4185:simplex
3788:objects
3761:inverse
3291:smaller
3252:coarser
3030:on the
2599:is the
2183:is the
2157:trivial
1821:if its
1552:subsets
722:(i.e.,
626:if the
202:Riemann
130:formula
120:History
6770:Topics
6572:metric
6447:Fields
6373:
6352:
6336:
6310:
6294:
6279:
6261:
6222:
6190:
6159:
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6107:
6048:
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5949:
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5764:
5754:
5524:graphs
5428:filter
5373:where
5182:. On
5178:or an
5102:, and
4929:Every
4924:metric
4402:under
4179:has a
4175:Every
4110:where
3963:is an
3813:, and
3786:whose
3299:weaker
3287:larger
3026:of an
2859:Using
2725:Given
2522:Given
2314:Given
2185:family
2102:Given
2071:[i.e.
1975:[i.e.
1787:closed
628:axioms
368:points
326:. Let
270:German
210:Jordan
206:Möbius
179:Cauchy
89:, and
83:limits
68:axioms
56:points
6552:Space
6105:S2CID
6075:(PDF)
6038:JSTOR
5832:
5615:up to
5540:of a
5532:edges
5526:with
5450:then
5426:is a
5071:balls
4351:is a
4225:split
3801:) by
3796:up to
3755:is a
3328:: if
3254:than
3193:than
3191:finer
1617:union
1523:on a
900:then
650:with
394:be a
6371:ISBN
6350:ISBN
6334:ISBN
6322:and
6308:ISBN
6292:ISBN
6277:ISBN
6259:ISBN
6220:ISBN
6188:ISBN
6157:ISBN
6136:ISBN
5947:ISBN
5873:2022
5860:ISBN
5762:OCLC
5752:ISBN
5530:and
5288:(or
5170:The
5163:Any
5156:and
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4702:The
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