6719:
1843:
6502:
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6708:
6777:
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186:, 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."
4715:
3817:(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
5457:
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
4770:
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.
4403:
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.
3916:
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
5429:. 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
1386:
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
2998:
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.
201:
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
2786:
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
5594:
3268:
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
2459:
1157:
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
3365:
2251:
221:
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
97:
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
6219:
4132:
4339:, 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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193:'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". "
3813:
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
2354:
269:
5680: – 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
59:
formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through
5546:
3285:
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.
5934:
3846:
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
3735:
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
6414:
6115:
5842:. Göschens Lehrbücherei/Gruppe I: Reine und Angewandte Mathematik Serie (in German). Leipzig: Von Veit (published 2011). p. 211.
5231:
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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217:" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by
4167:
are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
3315:
2173:
6758:
3751:
if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.
171:
4318:. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.
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75:
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5463:
5430:
4767:
4315:
3771:
3084:
607:
5633:– The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.
1866:
be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set
6753:
5811:
5507:
4758:
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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1806:
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32:
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36:
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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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Many topologies can be defined on a set to form a topological space. When every open set of a topology
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40:
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83:
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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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333:
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198:
175:
114:
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6252:
6092:
5659:
5645: – Generalization of the notion of convergence that is found in general topology
5636:
4898:
4734:
3782:
3736:
3732:
3438:
3391:
1991:] is missing; the bottom-right example is not a topology because the intersection of
245:
203:
6743:
6635:
6555:
6501:
5209:
4303:
is the simplest non-discrete topological space. It has important relations to the
4285:
4281:
4270:
3851:
3843:
3830:
163:
118:
87:
5837:
4766:, which is generated by the inverse images of open sets of the factors under the
3778:
are continuous functions. The attempt to classify the objects of this category (
6733:
6645:
6132:
6047:
5471:
4171:
3883:
210:
20:
6589:
6520:
6479:
6202:, (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997).
4851:
4643:
4311:
4274:
2849:
6614:
6215:
3740:
3015:
2584:
1551:
6139:(Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997).
5749:
5589:{\displaystyle \operatorname {cl} \{x\}\subseteq \operatorname {cl} \{y\},}
3277:
are sometimes used in place of finer and coarser, respectively. The terms
1183:
A standard example of such a system of neighbourhoods is for the real line
278:
is shown by the fact that there are several equivalent definitions of this
5861:
Unter einem m e t r i s c h e n R a u m e verstehen wir eine Menge
4672:
consists of the so-called "marked metric graph structures" of volume 1 on
2887:
Using these axioms, another way to define a topological space is as a set
6599:
6567:
6516:
6423:
4326:
4284:
has a natural topology that generalizes many of the geometric aspects of
4178:
3798:
3790:
3775:
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928:
288:
91:
60:
48:
28:
5729:
2454:{\displaystyle \tau =\{\varnothing ,\{2\},\{1,2\},\{2,3\},\{1,2,3\},X\}}
6165:
6084:
6025:
5142:
we construct a basis set consisting of all subsets of the union of the
4186:
3711:
This relates easily to the usual definition in analysis. Equivalently,
286:
suited for the application. The most commonly used is that in terms of
233:
first expressed the idea that a surface is a topological space that is
4044:
have a standard topology in which the basic open sets are open balls.
1535:
6017:
1694:
As this definition of a topology is the most commonly used, the set
4968:
is continuous. A common example of a quotient topology is when an
252:, though it was Hausdorff who popularised the term "metric space" (
5422:
3779:
1841:
611:
225:. His first article on this topic appeared in 1894. In the 1930s,
158:
relating the number of vertices (V), edges (E) and faces (F) of a
56:
2880:
The intersection of any collection of closed sets is also closed.
166:. The study and generalization of this formula, specifically by
6396:
1907:
The bottom-left example is not a topology because the union of
5689: – subset of a topological space whose closure is compact
4708:
3360:{\displaystyle F=\left\{\tau _{\alpha }:\alpha \in A\right\}}
2883:
The union of any finite number of closed sets is also closed.
2847:, the above axioms defining open sets become axioms defining
2246:{\displaystyle \tau =\{\{\},\{1,2,3,4\}\}=\{\varnothing ,X\}}
803:
721:
643:
591:
544:
487:
431:
363:
6392:
5026:
on the set of all non-empty subsets of a topological space
3051:. A topology is completely determined if for every net in
794:
belongs to every one of its neighbourhoods with respect to
35:
is defined but cannot necessarily be measured by a numeric
6221:
3002:
Another way to define a topological space is by using the
4346:. Here, the basic open sets are the half open intervals
5773:
5771:
4722:
It has been suggested that portions of this section be
3288:
The collection of all topologies on a given fixed set
5983:
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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
5682:
Pages displaying wikidata descriptions as a fallback
4762:
of topological spaces, the product can be given the
1648:
The intersection of any finite number of members of
6659:
6623:
6509:
6430:
6056:"Moduli of graphs and automorphisms of free groups"
3461:is the meet of the collection of all topologies on
244:in 1914 in his seminal "Principles of Set Theory".
6259:, Prentice Hall; 2nd edition (December 28, 1999).
5996:
5969:
5799:
5698: – Mathematical set with some added structure
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354:, though they can be any mathematical object. Let
342:
322:
150:
66:A topological space is the most general type of a
5053:, is generated by the following basis: for every
4185:since it is locally Euclidean. Similarly, every
292:, but perhaps more intuitive is that in terms of
63:, which is easier than the others to manipulate.
6006:Proceedings of the American Mathematical Society
5205:on the set of all non-empty closed subsets of a
4269:the closed sets of the Zariski topology are the
5668: – Semicontinuity for set-valued functions
5015:is then the natural projection onto the set of
3839:, a precise notion of distance between points.
5836:(1914) . "Punktmengen in allgemeinen Räumen".
5421:Topological spaces can be broadly classified,
3857:There are many ways of defining a topology on
2970:are the closed sets, and their complements in
47:, along with an additional structure called a
39:. More specifically, a topological space is a
6408:
5731:Introduction to metric and topological spaces
4310:There exist numerous topologies on any given
82:. Common types of topological spaces include
16:Mathematical space with a notion of closeness
8:
5580:
5574:
5562:
5556:
5169:that have non-empty intersections with each
4127:{\displaystyle \{\varnothing \}\cup \Gamma }
4115:
4109:
2543:
2519:
2448:
2439:
2421:
2415:
2403:
2397:
2385:
2379:
2373:
2364:
2335:
2311:
2240:
2228:
2222:
2219:
2195:
2189:
2186:
2183:
2123:
2099:
2068:
2062:
2042:
2030:
2010:
1998:
1978:
1966:
1946:
1940:
1920:
1914:
1891:
1873:
6241:, McGraw-Hill; 1st edition (June 1, 1968).
5449:Topological spaces with algebraic structure
3747:is also continuous. Two spaces are called
2760:of all finite subsets of the integers plus
2273:required by the axioms forms a topology on
1274:if it includes an open interval containing
330:be a (possibly empty) set. The elements of
270:Axiomatic foundations of topological spaces
6776:
6749:
6415:
6401:
6393:
5380:and have nonempty intersections with each
6231:General investigations of curved surfaces
6182:(3rd edition of differently titled books)
6155:Elements of Mathematics: General Topology
5988:
5982:
5961:
5955:
5950:Anderson, B. A.; Stewart, D. G. (1969). "
5912:
5900:
5789:J. Stillwell, Mathematics and its history
5777:
5601:
5548:
5522:
5391:
5385:
5365:
5345:
5322:
5302:
5281:
5262:
5256:
5236:
5216:
5180:
5174:
5153:
5147:
5124:
5103:
5084:
5078:
5058:
5031:
5000:
4977:
4953:
4933:
4906:
4882:
4862:
4823:
4803:
4783:
4683:
4677:
4656:
4650:
4619:
4599:
4579:
4547:
4512:
4478:
4436:
4412:
4389:
4388:
4386:
4351:
4250:
4246:
4245:
4242:
4221:
4217:
4216:
4213:
4139:
4107:
4087:
4063:
4028:
4024:
4023:
4020:
3994:
3993:
3991:
3966:
3962:
3961:
3958:
3933:
3929:
3928:
3925:
3912:. The set of all open intervals forms a
3894:
3893:
3891:
3865:
3864:
3862:
3854:this topology is the same for all norms.
3789:has motivated areas of research, such as
3716:
3678:
3658:
3638:
3609:
3589:
3563:
3525:
3486:
3466:
3446:
3419:
3399:
3372:
3334:
3317:
3293:
3249:
3243:
3218:
3212:
3188:
3182:
3157:
3151:
3127:
3121:
3100:
3094:
3056:
3023:
2975:
2955:
2932:
2912:
2892:
2862:
2817:
2795:
2794:
2792:
2768:
2767:
2765:
2745:
2723:
2722:
2714:
2673:
2650:
2615:
2592:
2568:
2511:
2486:
2466:
2356:
2303:
2278:
2258:
2175:
2152:
2091:
2060:
2028:
1996:
1964:
1938:
1912:
1871:
1851:
1813:
1778:
1748:
1723:
1699:
1673:
1653:
1628:
1608:
1579:
1559:
1519:
1472:
1452:
1432:
1412:
1392:
1368:
1348:
1324:
1304:
1279:
1259:
1236:
1235:
1233:
1213:
1191:
1190:
1188:
1163:
1136:
1116:
1096:
1076:
1056:
1036:
1011:
991:
962:
936:
909:
889:
866:
846:
826:
802:
801:
799:
779:
750:
720:
719:
711:
691:
671:
642:
641:
639:
619:
590:
589:
587:
566:
543:
542:
540:
520:
486:
485:
483:
460:
430:
429:
427:
407:
387:
362:
361:
359:
335:
315:
240:Topological spaces were first defined by
184:General investigations of curved surfaces
125:
6290:, Macdonald Technical & Scientific,
6274:, Springer; 1st edition (July 6, 2005).
5715:
5662: – Type of topological vector space
5708:
5674: – In mathematics, vector subspace
5478:Topological spaces with order structure
4112:
2367:
2231:
1818:
3047:is a generalisation of the concept of
3006:, which define the closed sets as the
2645:consisting of all possible subsets of
2253:consisting of only the two subsets of
1714:of the open sets is commonly called a
774:In other words, each point of the set
738:{\displaystyle N\in {\mathcal {N}}(x)}
6316:, Holt, Rinehart and Winston (1970).
5888:
5876:
5762:
3552:between topological spaces is called
1546:and satisfying the following axioms:
7:
6239:Schaum's Outline of General Topology
5820:participating institution membership
5411:Classification of topological spaces
4972:is defined on the topological space
4342:The real line can also be given the
1363:is a neighbourhood of all points in
1131:is a neighbourhood of each point of
248:had been defined earlier in 1906 by
5616:denotes an operator satisfying the
5609:{\displaystyle \operatorname {cl} }
5290:{\displaystyle U_{1},\ldots ,U_{n}}
5112:{\displaystyle U_{1},\ldots ,U_{n}}
4462:{\displaystyle \gamma =[0,\gamma )}
4193:inherits a natural topology from .
2668:In this case the topological space
1599:Any arbitrary (finite or infinite)
51:, which can be defined as a set of
6272:A Taste of Topology (Universitext)
6116:Undergraduate Texts in Mathematics
5639: – Type of mathematical space
4121:
4065:
2623:
209:The subject is clearly defined by
70:that allows for the definition of
14:
5656: – Type of topological space
4567:{\displaystyle (\alpha ,\gamma )}
4501:{\displaystyle (\alpha ,\beta ),}
4262:{\displaystyle \mathbb {C} ^{n},}
3982:the basic open sets are the open
3949:can be given a topology. In the
3774:are topological spaces and whose
3367:is a collection of topologies on
1299:Given such a structure, a subset
504:{\displaystyle {\mathcal {N}}(x)}
448:{\displaystyle {\mathcal {N}}(x)}
55:for each point that satisfy some
6775:
6748:
6738:
6728:
6717:
6707:
6706:
6500:
4877:is the collection of subsets of
4857:, then the quotient topology on
4713:
4230:{\displaystyle \mathbb {R} ^{n}}
4200:is defined algebraically on the
4037:{\displaystyle \mathbb {C} ^{n}}
3975:{\displaystyle \mathbb {R} ^{n}}
3942:{\displaystyle \mathbb {R} ^{n}}
3704:{\displaystyle f(M)\subseteq N.}
2740:the set of integers, the family
861:and includes a neighbourhood of
274:The utility of the concept of a
5492:if and only if it is the prime
5441:. For algebraic invariants see
3821:where limit points are unique.
2733:{\displaystyle X=\mathbb {Z} ,}
1514:may be defined as a collection
6229:Gauss, Carl Friedrich (1827).
4834:
4726:out into articles titled
4561:
4549:
4526:
4514:
4492:
4480:
4456:
4444:
4365:
4353:
4332:topology on any infinite set.
3805:Examples of topological spaces
3768:category of topological spaces
3689:
3683:
3620:
3614:
3536:
2950:Thus the sets in the topology
2687:
2675:
2632:
2626:
2552:{\displaystyle X=\{1,2,3,4\},}
2344:{\displaystyle X=\{1,2,3,4\},}
2132:{\displaystyle X=\{1,2,3,4\},}
1792:
1780:
931:of a neighbourhood of a point
811:{\displaystyle {\mathcal {N}}}
732:
726:
651:{\displaystyle {\mathcal {N}}}
614:below are satisfied; and then
599:{\displaystyle {\mathcal {N}}}
552:{\displaystyle {\mathcal {N}}}
498:
492:
442:
436:
371:{\displaystyle {\mathcal {N}}}
306:This axiomatization is due to
235:locally like a Euclidean plane
1:
6328:Vaidyanathaswamy, R. (1960).
5929:. Pearson. pp. 317–319.
4004:{\displaystyle \mathbb {C} ,}
3875:{\displaystyle \mathbb {R} ,}
3743:that is continuous and whose
3481:that contain every member of
2805:{\displaystyle \mathbb {Z} ,}
2638:{\displaystyle \tau =\wp (X)}
1201:{\displaystyle \mathbb {R} ,}
302:Definition via neighbourhoods
5734:. Oxford : Clarendon Press.
4396:{\displaystyle \mathbb {R} }
3901:{\displaystyle \mathbb {R} }
3886:. The standard topology on
3116:is also open for a topology
2775:{\displaystyle \mathbb {Z} }
1827:{\displaystyle X\setminus C}
1762:{\displaystyle C\subseteq X}
1243:{\displaystyle \mathbb {R} }
957:is again a neighbourhood of
298:and so this is given first.
6381:Encyclopedia of Mathematics
6313:Counterexamples in Topology
5687:Relatively compact subspace
4798:is a topological space and
4535:{\displaystyle [0,\beta ),}
4473:generated by the intervals
2907:together with a collection
6823:
6669:Banach fixed-point theorem
6110:Armstrong, M. A. (1983) .
5728:Sutherland, W. A. (1975).
5414:
5407:is a member of the basis.
5340:the set of all subsets of
5317:and for every compact set
4928:is the finest topology on
4778:is defined as follows: if
4051:
3850:. On a finite-dimensional
3828:
3510:
3261:{\displaystyle \tau _{2}.}
3200:{\displaystyle \tau _{1},}
3139:{\displaystyle \tau _{2},}
3082:
2839:Definition via closed sets
2481:forms another topology of
1900:{\displaystyle \{1,2,3\}.}
267:
227:James Waddell Alexander II
43:whose elements are called
6702:
6498:
6347:Willard, Stephen (2004).
6332:. Chelsea Publishing Co.
5925:Munkres, James R (2015).
5839:GrundzĂĽge der Mengenlehre
5807:Oxford English Dictionary
5618:Kuratowski closure axioms
5464:topological vector spaces
4705:Topological constructions
4335:Any set can be given the
4321:Any set can be given the
4316:finite topological spaces
4314:. Such spaces are called
3758:, one of the fundamental
3633:there is a neighbourhood
3227:{\displaystyle \tau _{1}}
3166:{\displaystyle \tau _{2}}
3109:{\displaystyle \tau _{1}}
3004:Kuratowski closure axioms
2693:{\displaystyle (X,\tau )}
1798:{\displaystyle (X,\tau )}
1499:Definition via open sets
1407:to be a neighbourhood of
1071:includes a neighbourhood
986:of two neighbourhoods of
6286:Schubert, Horst (1968),
6192:, Academic Press (1969).
6157:, Addison-Wesley (1966).
6064:Inventiones Mathematicae
5631:Complete Heyting algebra
5425:homeomorphism, by their
4843:{\displaystyle f:X\to Y}
4630:{\displaystyle \gamma .}
4469:may be endowed with the
3584:and every neighbourhood
3545:{\displaystyle f:X\to Y}
3085:Comparison of topologies
3079:Comparison of topologies
27:is, roughly speaking, a
5812:Oxford University Press
5536:{\displaystyle x\leq y}
5509:specialization preorder
5504:Specialization preorder
5360:that are disjoint from
4587:{\displaystyle \alpha }
4420:{\displaystyle \gamma }
4071:{\displaystyle \Gamma }
3835:Metric spaces embody a
3414:is the intersection of
2812:and so it cannot be in
2048:{\displaystyle \{2,3\}}
2016:{\displaystyle \{1,2\}}
1984:{\displaystyle \{2,3\}}
1489:{\displaystyle x\in U.}
767:{\displaystyle x\in N.}
422:a non-empty collection
282:. Thus one chooses the
219:Johann Benedict Listing
151:{\displaystyle V-E+F=2}
6724:Mathematics portal
6624:Metrics and properties
6610:Second-countable space
6353:. Dover Publications.
6308:Seebach, J. Arthur Jr.
6167:Topology and Groupoids
5998:
5971:
5678:Quasitopological space
5610:
5590:
5537:
5427:topological properties
5401:
5374:
5354:
5334:
5311:
5291:
5245:
5225:
5193:
5192:{\displaystyle U_{i}.}
5163:
5136:
5113:
5067:
5043:
5009:
4989:
4962:
4942:
4918:
4891:
4871:
4844:
4812:
4792:
4696:
4695:{\displaystyle F_{n}.}
4666:
4631:
4608:
4607:{\displaystyle \beta }
4588:
4568:
4536:
4502:
4463:
4421:
4397:
4375:
4374:{\displaystyle [a,b).}
4263:
4231:
4151:
4128:
4096:
4072:
4038:
4005:
3976:
3943:
3902:
3876:
3725:
3705:
3667:
3647:
3627:
3598:
3578:
3577:{\displaystyle x\in X}
3546:
3498:
3475:
3455:
3431:
3408:
3384:
3361:
3302:
3262:
3228:
3201:
3167:
3140:
3110:
3065:
3035:
2984:
2964:
2944:
2921:
2901:
2871:
2829:
2828:{\displaystyle \tau .}
2806:
2776:
2754:
2734:
2694:
2662:
2639:
2604:
2577:
2553:
2498:
2475:
2455:
2345:
2290:
2267:
2247:
2161:
2133:
2082:
2075:
2049:
2017:
1985:
1953:
1927:
1901:
1860:
1838:Examples of topologies
1828:
1799:
1763:
1735:
1708:
1685:
1684:{\displaystyle \tau .}
1662:
1640:
1639:{\displaystyle \tau .}
1617:
1591:
1590:{\displaystyle \tau .}
1568:
1528:
1490:
1461:
1441:
1421:
1401:
1380:
1357:
1333:
1313:
1291:
1268:
1244:
1222:
1202:
1175:
1148:
1125:
1105:
1085:
1065:
1045:
1023:
1006:is a neighbourhood of
1000:
974:
951:
950:{\displaystyle x\in X}
921:
904:is a neighbourhood of
898:
878:
855:
835:
812:
788:
768:
739:
700:
686:is a neighbourhood of
680:
652:
628:
608:neighbourhood topology
600:
575:
553:
529:
505:
472:
449:
416:
396:
372:
344:
324:
280:mathematical structure
257:
178:of topology. In 1827,
152:
6137:Topology and Geometry
5999:
5997:{\displaystyle T_{1}}
5972:
5970:{\displaystyle T_{1}}
5778:Gallier & Xu 2013
5611:
5591:
5538:
5402:
5400:{\displaystyle U_{i}}
5375:
5355:
5335:
5312:
5292:
5246:
5226:
5194:
5164:
5162:{\displaystyle U_{i}}
5137:
5114:
5068:
5044:
5010:
4990:
4963:
4943:
4919:
4892:
4872:
4845:
4813:
4793:
4697:
4667:
4665:{\displaystyle F_{n}}
4632:
4609:
4589:
4569:
4537:
4503:
4464:
4422:
4398:
4376:
4305:theory of computation
4264:
4232:
4152:
4129:
4097:
4073:
4039:
4006:
3977:
3944:
3903:
3877:
3731:is continuous if the
3726:
3706:
3668:
3648:
3628:
3599:
3579:
3547:
3499:
3476:
3456:
3432:
3409:
3385:
3362:
3303:
3263:
3229:
3202:
3168:
3141:
3111:
3066:
3036:
2985:
2965:
2963:{\displaystyle \tau }
2945:
2927:of closed subsets of
2922:
2920:{\displaystyle \tau }
2902:
2872:
2830:
2807:
2777:
2755:
2753:{\displaystyle \tau }
2735:
2695:
2663:
2640:
2605:
2578:
2554:
2499:
2476:
2456:
2346:
2291:
2268:
2248:
2162:
2134:
2076:
2074:{\displaystyle \{2\}}
2050:
2018:
1986:
1954:
1952:{\displaystyle \{3\}}
1928:
1926:{\displaystyle \{2\}}
1902:
1861:
1859:{\displaystyle \tau }
1845:
1829:
1800:
1764:
1736:
1709:
1707:{\displaystyle \tau }
1686:
1663:
1661:{\displaystyle \tau }
1641:
1618:
1616:{\displaystyle \tau }
1592:
1569:
1529:
1527:{\displaystyle \tau }
1491:
1462:
1447:includes an open set
1442:
1422:
1402:
1381:
1358:
1334:
1314:
1292:
1269:
1245:
1223:
1203:
1176:
1149:
1126:
1106:
1086:
1066:
1046:
1024:
1001:
975:
952:
922:
899:
879:
856:
836:
813:
789:
769:
740:
701:
681:
653:
629:
601:
576:
554:
530:
506:
473:
450:
417:
397:
373:
345:
325:
153:
6679:Invariance of domain
6631:Euler characteristic
6605:Bundle (mathematics)
6237:Lipschutz, Seymour;
5981:
5954:
5600:
5547:
5521:
5417:Topological property
5384:
5364:
5344:
5321:
5301:
5255:
5235:
5215:
5173:
5146:
5123:
5077:
5057:
5030:
4999:
4976:
4970:equivalence relation
4952:
4932:
4924:In other words, the
4905:
4881:
4861:
4822:
4802:
4782:
4676:
4649:
4618:
4598:
4578:
4546:
4511:
4477:
4435:
4411:
4385:
4350:
4344:lower limit topology
4337:cocountable topology
4241:
4212:
4138:
4106:
4086:
4062:
4019:
3990:
3957:
3924:
3908:is generated by the
3890:
3861:
3766:, which denotes the
3715:
3677:
3657:
3637:
3626:{\displaystyle f(x)}
3608:
3588:
3562:
3524:
3507:Continuous functions
3485:
3465:
3445:
3418:
3398:
3371:
3316:
3292:
3242:
3211:
3181:
3150:
3120:
3093:
3055:
3022:
2974:
2954:
2931:
2911:
2891:
2861:
2816:
2791:
2764:
2744:
2713:
2672:
2649:
2614:
2610:which is the family
2591:
2567:
2510:
2485:
2465:
2355:
2302:
2277:
2257:
2174:
2151:
2090:
2059:
2027:
1995:
1963:
1937:
1911:
1870:
1850:
1812:
1777:
1747:
1722:
1698:
1672:
1652:
1627:
1607:
1578:
1558:
1518:
1471:
1451:
1431:
1411:
1391:
1367:
1347:
1323:
1303:
1278:
1258:
1232:
1212:
1187:
1162:
1135:
1115:
1095:
1075:
1055:
1035:
1010:
990:
961:
935:
908:
888:
865:
845:
825:
798:
778:
749:
710:
690:
670:
638:
618:
586:
565:
539:
519:
482:
459:
426:
406:
386:
358:
334:
314:
180:Carl Friedrich Gauss
124:
6689:Tychonoff's theorem
6684:Poincaré conjecture
6438:General (point-set)
6376:"Topological space"
6077:1986InMat..84...91C
5810:(Online ed.).
5696:Space (mathematics)
5017:equivalence classes
4165:functional analysis
3848:normed vector space
3513:Continuous function
3073:accumulation points
2990:are the open sets.
1250:is defined to be a
350:are usually called
6807:Topological spaces
6674:De Rham cohomology
6595:Polyhedral complex
6585:Simplicial complex
6200:Algebraic Topology
6085:10.1007/BF01388734
5994:
5967:
5606:
5586:
5533:
5513:canonical preorder
5506:: In a space the
5494:spectrum of a ring
5460:topological groups
5443:algebraic topology
5397:
5370:
5350:
5333:{\displaystyle K,}
5330:
5307:
5287:
5241:
5221:
5189:
5159:
5135:{\displaystyle X,}
5132:
5109:
5063:
5042:{\displaystyle X,}
5039:
5005:
4988:{\displaystyle X.}
4985:
4958:
4938:
4917:{\displaystyle f.}
4914:
4887:
4867:
4840:
4808:
4788:
4692:
4662:
4627:
4604:
4584:
4564:
4532:
4498:
4459:
4417:
4393:
4371:
4259:
4227:
4202:spectrum of a ring
4191:simplicial complex
4150:{\displaystyle X.}
4147:
4124:
4092:
4068:
4054:List of topologies
4034:
4001:
3972:
3939:
3898:
3872:
3721:
3701:
3663:
3643:
3623:
3594:
3574:
3542:
3497:{\displaystyle F.}
3494:
3471:
3451:
3430:{\displaystyle F,}
3427:
3404:
3383:{\displaystyle X,}
3380:
3357:
3298:
3258:
3224:
3197:
3163:
3136:
3106:
3061:
3034:{\displaystyle X.}
3031:
2980:
2960:
2943:{\displaystyle X.}
2940:
2917:
2897:
2867:
2857:The empty set and
2825:
2802:
2772:
2750:
2730:
2690:
2661:{\displaystyle X.}
2658:
2635:
2603:{\displaystyle X,}
2600:
2573:
2549:
2497:{\displaystyle X.}
2494:
2471:
2461:of six subsets of
2451:
2341:
2289:{\displaystyle X.}
2286:
2263:
2243:
2157:
2129:
2083:
2071:
2045:
2013:
1981:
1949:
1923:
1897:
1856:
1824:
1795:
1759:
1734:{\displaystyle X.}
1731:
1704:
1681:
1658:
1636:
1613:
1587:
1564:
1524:
1486:
1457:
1437:
1417:
1397:
1379:{\displaystyle U.}
1376:
1353:
1329:
1309:
1290:{\displaystyle x.}
1287:
1264:
1240:
1218:
1198:
1174:{\displaystyle X.}
1171:
1147:{\displaystyle M.}
1144:
1121:
1101:
1081:
1061:
1041:
1031:Any neighbourhood
1022:{\displaystyle x.}
1019:
996:
973:{\displaystyle x.}
970:
947:
920:{\displaystyle x.}
917:
894:
877:{\displaystyle x,}
874:
851:
831:
808:
784:
764:
735:
696:
676:
648:
624:
596:
571:
561:neighbourhoods of
549:
525:
501:
471:{\displaystyle X.}
468:
445:
412:
392:
382:assigning to each
368:
340:
320:
148:
99:point-set topology
68:mathematical space
6789:
6788:
6578:fundamental group
6151:Bourbaki, Nicolas
5936:978-93-325-4953-1
5903:, definition 2.1.
5818:(Subscription or
5643:Convergence space
5468:topological rings
5458:concepts such as
5455:algebraic objects
5439:separation axioms
5373:{\displaystyle K}
5353:{\displaystyle X}
5310:{\displaystyle X}
5244:{\displaystyle n}
5224:{\displaystyle X}
5066:{\displaystyle n}
5024:Vietoris topology
5008:{\displaystyle f}
4961:{\displaystyle f}
4941:{\displaystyle Y}
4926:quotient topology
4890:{\displaystyle Y}
4870:{\displaystyle Y}
4818:is a set, and if
4811:{\displaystyle Y}
4791:{\displaystyle X}
4756:subspace topology
4752:
4751:
4747:
4729:Vietoris topology
4381:This topology on
4323:cofinite topology
4206:algebraic variety
4134:is a topology on
4095:{\displaystyle X}
3811:discrete topology
3724:{\displaystyle f}
3666:{\displaystyle x}
3646:{\displaystyle M}
3597:{\displaystyle N}
3474:{\displaystyle X}
3454:{\displaystyle F}
3407:{\displaystyle F}
3301:{\displaystyle X}
3064:{\displaystyle X}
2994:Other definitions
2983:{\displaystyle X}
2900:{\displaystyle X}
2870:{\displaystyle X}
2576:{\displaystyle X}
2561:discrete topology
2474:{\displaystyle X}
2266:{\displaystyle X}
2160:{\displaystyle X}
1574:itself belong to
1567:{\displaystyle X}
1460:{\displaystyle U}
1440:{\displaystyle N}
1420:{\displaystyle x}
1400:{\displaystyle N}
1356:{\displaystyle U}
1339:is defined to be
1332:{\displaystyle X}
1312:{\displaystyle U}
1267:{\displaystyle x}
1254:of a real number
1221:{\displaystyle N}
1124:{\displaystyle N}
1104:{\displaystyle x}
1084:{\displaystyle M}
1064:{\displaystyle x}
1044:{\displaystyle N}
999:{\displaystyle x}
897:{\displaystyle N}
854:{\displaystyle X}
834:{\displaystyle N}
787:{\displaystyle X}
699:{\displaystyle x}
679:{\displaystyle N}
660:topological space
627:{\displaystyle X}
574:{\displaystyle x}
528:{\displaystyle x}
415:{\displaystyle X}
395:{\displaystyle x}
343:{\displaystyle X}
323:{\displaystyle X}
176:boosted the study
162:, and hence of a
160:convex polyhedron
29:geometrical space
25:topological space
6814:
6802:General topology
6779:
6778:
6752:
6751:
6742:
6732:
6722:
6721:
6710:
6709:
6504:
6417:
6410:
6403:
6394:
6389:
6364:
6350:General Topology
6343:
6300:
6234:
6225:
6181:
6129:
6097:
6096:
6060:
6044:
6038:
6037:
6003:
6001:
6000:
5995:
5993:
5992:
5977:-complements of
5976:
5974:
5973:
5968:
5966:
5965:
5947:
5941:
5940:
5922:
5916:
5910:
5904:
5898:
5892:
5886:
5880:
5874:
5868:
5867:
5858:
5856:
5834:Hausdorff, Felix
5830:
5824:
5823:
5815:
5803:
5796:
5790:
5787:
5781:
5775:
5766:
5760:
5754:
5753:
5725:
5719:
5713:
5692:
5683:
5615:
5613:
5612:
5607:
5595:
5593:
5592:
5587:
5542:
5540:
5539:
5534:
5406:
5404:
5403:
5398:
5396:
5395:
5379:
5377:
5376:
5371:
5359:
5357:
5356:
5351:
5339:
5337:
5336:
5331:
5316:
5314:
5313:
5308:
5297:of open sets in
5296:
5294:
5293:
5288:
5286:
5285:
5267:
5266:
5250:
5248:
5247:
5242:
5230:
5228:
5227:
5222:
5198:
5196:
5195:
5190:
5185:
5184:
5168:
5166:
5165:
5160:
5158:
5157:
5141:
5139:
5138:
5133:
5119:of open sets in
5118:
5116:
5115:
5110:
5108:
5107:
5089:
5088:
5072:
5070:
5069:
5064:
5051:Leopold Vietoris
5048:
5046:
5045:
5040:
5014:
5012:
5011:
5006:
4994:
4992:
4991:
4986:
4967:
4965:
4964:
4959:
4947:
4945:
4944:
4939:
4923:
4921:
4920:
4915:
4896:
4894:
4893:
4888:
4876:
4874:
4873:
4868:
4849:
4847:
4846:
4841:
4817:
4815:
4814:
4809:
4797:
4795:
4794:
4789:
4764:product topology
4743:
4717:
4716:
4709:
4701:
4699:
4698:
4693:
4688:
4687:
4671:
4669:
4668:
4663:
4661:
4660:
4636:
4634:
4633:
4628:
4614:are elements of
4613:
4611:
4610:
4605:
4593:
4591:
4590:
4585:
4573:
4571:
4570:
4565:
4541:
4539:
4538:
4533:
4507:
4505:
4504:
4499:
4468:
4466:
4465:
4460:
4426:
4424:
4423:
4418:
4402:
4400:
4399:
4394:
4392:
4380:
4378:
4377:
4372:
4301:Sierpiński space
4268:
4266:
4265:
4260:
4255:
4254:
4249:
4236:
4234:
4233:
4228:
4226:
4225:
4220:
4198:Zariski topology
4183:natural topology
4161:linear operators
4156:
4154:
4153:
4148:
4133:
4131:
4130:
4125:
4101:
4099:
4098:
4093:
4077:
4075:
4074:
4069:
4043:
4041:
4040:
4035:
4033:
4032:
4027:
4010:
4008:
4007:
4002:
3997:
3981:
3979:
3978:
3973:
3971:
3970:
3965:
3948:
3946:
3945:
3940:
3938:
3937:
3932:
3919:Euclidean spaces
3907:
3905:
3904:
3899:
3897:
3881:
3879:
3878:
3873:
3868:
3819:Hausdorff spaces
3815:trivial topology
3730:
3728:
3727:
3722:
3710:
3708:
3707:
3702:
3672:
3670:
3669:
3664:
3652:
3650:
3649:
3644:
3632:
3630:
3629:
3624:
3603:
3601:
3600:
3595:
3583:
3581:
3580:
3575:
3551:
3549:
3548:
3543:
3503:
3501:
3500:
3495:
3480:
3478:
3477:
3472:
3460:
3458:
3457:
3452:
3436:
3434:
3433:
3428:
3413:
3411:
3410:
3405:
3389:
3387:
3386:
3381:
3366:
3364:
3363:
3358:
3356:
3352:
3339:
3338:
3310:complete lattice
3307:
3305:
3304:
3299:
3267:
3265:
3264:
3259:
3254:
3253:
3233:
3231:
3230:
3225:
3223:
3222:
3206:
3204:
3203:
3198:
3193:
3192:
3172:
3170:
3169:
3164:
3162:
3161:
3145:
3143:
3142:
3137:
3132:
3131:
3115:
3113:
3112:
3107:
3105:
3104:
3070:
3068:
3067:
3062:
3040:
3038:
3037:
3032:
2989:
2987:
2986:
2981:
2969:
2967:
2966:
2961:
2949:
2947:
2946:
2941:
2926:
2924:
2923:
2918:
2906:
2904:
2903:
2898:
2876:
2874:
2873:
2868:
2845:de Morgan's laws
2834:
2832:
2831:
2826:
2811:
2809:
2808:
2803:
2798:
2781:
2779:
2778:
2773:
2771:
2759:
2757:
2756:
2751:
2739:
2737:
2736:
2731:
2726:
2699:
2697:
2696:
2691:
2667:
2665:
2664:
2659:
2644:
2642:
2641:
2636:
2609:
2607:
2606:
2601:
2582:
2580:
2579:
2574:
2558:
2556:
2555:
2550:
2503:
2501:
2500:
2495:
2480:
2478:
2477:
2472:
2460:
2458:
2457:
2452:
2350:
2348:
2347:
2342:
2295:
2293:
2292:
2287:
2272:
2270:
2269:
2264:
2252:
2250:
2249:
2244:
2166:
2164:
2163:
2158:
2138:
2136:
2135:
2130:
2080:
2078:
2077:
2072:
2054:
2052:
2051:
2046:
2022:
2020:
2019:
2014:
1990:
1988:
1987:
1982:
1958:
1956:
1955:
1950:
1932:
1930:
1929:
1924:
1906:
1904:
1903:
1898:
1865:
1863:
1862:
1857:
1834:is an open set.
1833:
1831:
1830:
1825:
1804:
1802:
1801:
1796:
1768:
1766:
1765:
1760:
1740:
1738:
1737:
1732:
1713:
1711:
1710:
1705:
1690:
1688:
1687:
1682:
1667:
1665:
1664:
1659:
1645:
1643:
1642:
1637:
1622:
1620:
1619:
1614:
1596:
1594:
1593:
1588:
1573:
1571:
1570:
1565:
1541:
1533:
1531:
1530:
1525:
1513:
1495:
1493:
1492:
1487:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1426:
1424:
1423:
1418:
1406:
1404:
1403:
1398:
1385:
1383:
1382:
1377:
1362:
1360:
1359:
1354:
1338:
1336:
1335:
1330:
1318:
1316:
1315:
1310:
1296:
1294:
1293:
1288:
1273:
1271:
1270:
1265:
1249:
1247:
1246:
1241:
1239:
1227:
1225:
1224:
1219:
1207:
1205:
1204:
1199:
1194:
1180:
1178:
1177:
1172:
1153:
1151:
1150:
1145:
1130:
1128:
1127:
1122:
1110:
1108:
1107:
1102:
1090:
1088:
1087:
1082:
1070:
1068:
1067:
1062:
1050:
1048:
1047:
1042:
1028:
1026:
1025:
1020:
1005:
1003:
1002:
997:
979:
977:
976:
971:
956:
954:
953:
948:
926:
924:
923:
918:
903:
901:
900:
895:
883:
881:
880:
875:
860:
858:
857:
852:
840:
838:
837:
832:
817:
815:
814:
809:
807:
806:
793:
791:
790:
785:
773:
771:
770:
765:
744:
742:
741:
736:
725:
724:
705:
703:
702:
697:
685:
683:
682:
677:
657:
655:
654:
649:
647:
646:
633:
631:
630:
625:
605:
603:
602:
597:
595:
594:
582:). The function
580:
578:
577:
572:
558:
556:
555:
550:
548:
547:
535:with respect to
534:
532:
531:
526:
510:
508:
507:
502:
491:
490:
478:The elements of
477:
475:
474:
469:
454:
452:
451:
446:
435:
434:
421:
419:
418:
413:
401:
399:
398:
393:
377:
375:
374:
369:
367:
366:
349:
347:
346:
341:
329:
327:
326:
321:
215:Erlangen Program
170:(1789–1857) and
157:
155:
154:
149:
103:general topology
84:Euclidean spaces
6822:
6821:
6817:
6816:
6815:
6813:
6812:
6811:
6792:
6791:
6790:
6785:
6716:
6698:
6694:Urysohn's lemma
6655:
6619:
6505:
6496:
6468:low-dimensional
6426:
6421:
6374:
6371:
6361:
6346:
6340:
6327:
6298:
6285:
6270:Runde, Volker;
6228:
6214:Gallier, Jean;
6213:
6196:Fulton, William
6178:
6160:
6133:Bredon, Glen E.
6126:
6109:
6106:
6101:
6100:
6058:
6052:Vogtmann, Karen
6046:
6045:
6041:
6018:10.2307/2037491
5984:
5979:
5978:
5957:
5952:
5951:
5949:
5948:
5944:
5937:
5924:
5923:
5919:
5911:
5907:
5899:
5895:
5887:
5883:
5875:
5871:
5854:
5852:
5850:
5832:
5831:
5827:
5817:
5798:
5797:
5793:
5788:
5784:
5776:
5769:
5761:
5757:
5742:
5727:
5726:
5722:
5714:
5710:
5705:
5690:
5681:
5672:Linear subspace
5654:Hausdorff space
5627:
5598:
5597:
5545:
5544:
5543:if and only if
5519:
5518:
5480:
5451:
5419:
5413:
5387:
5382:
5381:
5362:
5361:
5342:
5341:
5319:
5318:
5299:
5298:
5277:
5258:
5253:
5252:
5233:
5232:
5213:
5212:
5207:locally compact
5176:
5171:
5170:
5149:
5144:
5143:
5121:
5120:
5099:
5080:
5075:
5074:
5055:
5054:
5028:
5027:
4997:
4996:
4974:
4973:
4950:
4949:
4930:
4929:
4903:
4902:
4897:that have open
4879:
4878:
4859:
4858:
4820:
4819:
4800:
4799:
4780:
4779:
4748:
4718:
4714:
4707:
4679:
4674:
4673:
4652:
4647:
4646:
4616:
4615:
4596:
4595:
4576:
4575:
4544:
4543:
4509:
4508:
4475:
4474:
4433:
4432:
4431:, then the set
4409:
4408:
4383:
4382:
4348:
4347:
4330:
4307:and semantics.
4244:
4239:
4238:
4215:
4210:
4209:
4136:
4135:
4104:
4103:
4084:
4083:
4060:
4059:
4056:
4050:
4022:
4017:
4016:
4013:complex numbers
3988:
3987:
3960:
3955:
3954:
3927:
3922:
3921:
3888:
3887:
3859:
3858:
3833:
3827:
3807:
3795:homology theory
3791:homotopy theory
3756:category theory
3713:
3712:
3675:
3674:
3655:
3654:
3635:
3634:
3606:
3605:
3586:
3585:
3560:
3559:
3522:
3521:
3515:
3509:
3483:
3482:
3463:
3462:
3443:
3442:
3416:
3415:
3396:
3395:
3369:
3368:
3330:
3329:
3325:
3314:
3313:
3290:
3289:
3245:
3240:
3239:
3214:
3209:
3208:
3184:
3179:
3178:
3153:
3148:
3147:
3123:
3118:
3117:
3096:
3091:
3090:
3087:
3081:
3071:the set of its
3053:
3052:
3020:
3019:
2996:
2972:
2971:
2952:
2951:
2929:
2928:
2909:
2908:
2889:
2888:
2859:
2858:
2841:
2814:
2813:
2789:
2788:
2762:
2761:
2742:
2741:
2711:
2710:
2670:
2669:
2647:
2646:
2612:
2611:
2589:
2588:
2565:
2564:
2508:
2507:
2483:
2482:
2463:
2462:
2353:
2352:
2300:
2299:
2275:
2274:
2255:
2254:
2172:
2171:
2149:
2148:
2088:
2087:
2057:
2056:
2025:
2024:
1993:
1992:
1961:
1960:
1935:
1934:
1909:
1908:
1868:
1867:
1848:
1847:
1840:
1810:
1809:
1775:
1774:
1745:
1744:
1720:
1719:
1696:
1695:
1670:
1669:
1650:
1649:
1625:
1624:
1605:
1604:
1576:
1575:
1556:
1555:
1539:
1516:
1515:
1511:
1501:
1469:
1468:
1449:
1448:
1429:
1428:
1409:
1408:
1389:
1388:
1365:
1364:
1345:
1344:
1321:
1320:
1301:
1300:
1276:
1275:
1256:
1255:
1230:
1229:
1210:
1209:
1208:where a subset
1185:
1184:
1160:
1159:
1133:
1132:
1113:
1112:
1093:
1092:
1073:
1072:
1053:
1052:
1033:
1032:
1008:
1007:
988:
987:
959:
958:
933:
932:
906:
905:
886:
885:
863:
862:
843:
842:
841:is a subset of
823:
822:
796:
795:
776:
775:
747:
746:
708:
707:
688:
687:
668:
667:
636:
635:
616:
615:
584:
583:
563:
562:
537:
536:
517:
516:
511:will be called
480:
479:
457:
456:
424:
423:
404:
403:
384:
383:
356:
355:
332:
331:
312:
311:
308:Felix Hausdorff
304:
272:
266:
258:metrischer Raum
250:Maurice Fréchet
242:Felix Hausdorff
231:Hassler Whitney
122:
121:
117:discovered the
111:
17:
12:
11:
5:
6820:
6818:
6810:
6809:
6804:
6794:
6793:
6787:
6786:
6784:
6783:
6773:
6772:
6771:
6766:
6761:
6746:
6736:
6726:
6714:
6703:
6700:
6699:
6697:
6696:
6691:
6686:
6681:
6676:
6671:
6665:
6663:
6657:
6656:
6654:
6653:
6648:
6643:
6641:Winding number
6638:
6633:
6627:
6625:
6621:
6620:
6618:
6617:
6612:
6607:
6602:
6597:
6592:
6587:
6582:
6581:
6580:
6575:
6573:homotopy group
6565:
6564:
6563:
6558:
6553:
6548:
6543:
6533:
6528:
6523:
6513:
6511:
6507:
6506:
6499:
6497:
6495:
6494:
6489:
6484:
6483:
6482:
6472:
6471:
6470:
6460:
6455:
6450:
6445:
6440:
6434:
6432:
6428:
6427:
6422:
6420:
6419:
6412:
6405:
6397:
6391:
6390:
6370:
6369:External links
6367:
6366:
6365:
6359:
6344:
6338:
6325:
6304:Steen, Lynn A.
6301:
6296:
6283:
6268:
6253:Munkres, James
6250:
6235:
6226:
6211:
6193:
6183:
6176:
6158:
6148:
6130:
6124:
6112:Basic Topology
6105:
6102:
6099:
6098:
6039:
5991:
5987:
5964:
5960:
5942:
5935:
5917:
5915:, theorem 2.6.
5913:Armstrong 1983
5905:
5901:Armstrong 1983
5893:
5891:, section 2.2.
5881:
5879:, section 2.1.
5869:
5848:
5825:
5801:"metric space"
5791:
5782:
5767:
5755:
5740:
5720:
5707:
5706:
5704:
5701:
5700:
5699:
5693:
5684:
5675:
5669:
5666:Hemicontinuity
5663:
5657:
5651:
5649:Exterior space
5646:
5640:
5634:
5626:
5623:
5622:
5621:
5605:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5561:
5558:
5555:
5552:
5532:
5529:
5526:
5517:is defined by
5501:
5479:
5476:
5450:
5447:
5437:, and various
5415:Main article:
5412:
5409:
5394:
5390:
5369:
5349:
5329:
5326:
5306:
5284:
5280:
5276:
5273:
5270:
5265:
5261:
5240:
5220:
5188:
5183:
5179:
5156:
5152:
5131:
5128:
5106:
5102:
5098:
5095:
5092:
5087:
5083:
5062:
5038:
5035:
5004:
4984:
4981:
4957:
4937:
4913:
4910:
4899:inverse images
4886:
4866:
4839:
4836:
4833:
4830:
4827:
4807:
4787:
4776:quotient space
4760:indexed family
4750:
4749:
4721:
4719:
4712:
4706:
4703:
4691:
4686:
4682:
4659:
4655:
4626:
4623:
4603:
4583:
4563:
4560:
4557:
4554:
4551:
4531:
4528:
4525:
4522:
4519:
4516:
4497:
4494:
4491:
4488:
4485:
4482:
4471:order topology
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4429:ordinal number
4416:
4391:
4370:
4367:
4364:
4361:
4358:
4355:
4328:
4273:of systems of
4258:
4253:
4248:
4224:
4219:
4146:
4143:
4123:
4120:
4117:
4114:
4111:
4091:
4067:
4049:
4046:
4031:
4026:
4000:
3996:
3986:. Similarly,
3969:
3964:
3951:usual topology
3936:
3931:
3910:open intervals
3896:
3871:
3867:
3829:Main article:
3826:
3823:
3806:
3803:
3750:
3720:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3662:
3642:
3622:
3619:
3616:
3613:
3593:
3573:
3570:
3567:
3541:
3538:
3535:
3532:
3529:
3511:Main article:
3508:
3505:
3493:
3490:
3470:
3450:
3426:
3423:
3403:
3379:
3376:
3355:
3351:
3348:
3345:
3342:
3337:
3333:
3328:
3324:
3321:
3297:
3284:
3280:
3276:
3272:
3257:
3252:
3248:
3237:
3221:
3217:
3196:
3191:
3187:
3176:
3160:
3156:
3146:one says that
3135:
3130:
3126:
3103:
3099:
3083:Main article:
3080:
3077:
3075:is specified.
3060:
3030:
3027:
2995:
2992:
2979:
2959:
2939:
2936:
2916:
2896:
2885:
2884:
2881:
2878:
2866:
2840:
2837:
2836:
2835:
2824:
2821:
2801:
2797:
2785:
2770:
2749:
2729:
2725:
2721:
2718:
2707:
2703:discrete space
2689:
2686:
2683:
2680:
2677:
2657:
2654:
2634:
2631:
2628:
2625:
2622:
2619:
2599:
2596:
2572:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2504:
2493:
2490:
2470:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2296:
2285:
2282:
2262:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2156:
2146:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2081:], is missing.
2070:
2067:
2064:
2044:
2041:
2038:
2035:
2032:
2012:
2009:
2006:
2003:
2000:
1980:
1977:
1974:
1971:
1968:
1948:
1945:
1942:
1922:
1919:
1916:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1855:
1839:
1836:
1823:
1820:
1817:
1794:
1791:
1788:
1785:
1782:
1772:
1769:is said to be
1758:
1755:
1752:
1730:
1727:
1703:
1692:
1691:
1680:
1677:
1657:
1646:
1635:
1632:
1612:
1603:of members of
1597:
1586:
1583:
1563:
1523:
1500:
1497:
1485:
1482:
1479:
1476:
1456:
1436:
1416:
1396:
1375:
1372:
1352:
1328:
1308:
1286:
1283:
1263:
1253:
1238:
1217:
1197:
1193:
1170:
1167:
1155:
1154:
1143:
1140:
1120:
1100:
1080:
1060:
1040:
1029:
1018:
1015:
995:
980:
969:
966:
946:
943:
940:
916:
913:
893:
873:
870:
850:
830:
819:
805:
783:
763:
760:
757:
754:
734:
731:
728:
723:
718:
715:
695:
675:
645:
623:
593:
581:
570:
546:
524:
514:
513:neighbourhoods
500:
497:
494:
489:
467:
464:
455:of subsets of
444:
441:
438:
433:
411:
391:
365:
353:
339:
319:
303:
300:
297:
295:neighbourhoods
291:
284:axiomatization
268:Main article:
265:
262:
223:Henri Poincaré
147:
144:
141:
138:
135:
132:
129:
115:Leonhard Euler
110:
107:
53:neighbourhoods
15:
13:
10:
9:
6:
4:
3:
2:
6819:
6808:
6805:
6803:
6800:
6799:
6797:
6782:
6774:
6770:
6767:
6765:
6762:
6760:
6757:
6756:
6755:
6747:
6745:
6741:
6737:
6735:
6731:
6727:
6725:
6720:
6715:
6713:
6705:
6704:
6701:
6695:
6692:
6690:
6687:
6685:
6682:
6680:
6677:
6675:
6672:
6670:
6667:
6666:
6664:
6662:
6658:
6652:
6651:Orientability
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6628:
6626:
6622:
6616:
6613:
6611:
6608:
6606:
6603:
6601:
6598:
6596:
6593:
6591:
6588:
6586:
6583:
6579:
6576:
6574:
6571:
6570:
6569:
6566:
6562:
6559:
6557:
6554:
6552:
6549:
6547:
6544:
6542:
6539:
6538:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6518:
6515:
6514:
6512:
6508:
6503:
6493:
6490:
6488:
6487:Set-theoretic
6485:
6481:
6478:
6477:
6476:
6473:
6469:
6466:
6465:
6464:
6461:
6459:
6456:
6454:
6451:
6449:
6448:Combinatorial
6446:
6444:
6441:
6439:
6436:
6435:
6433:
6429:
6425:
6418:
6413:
6411:
6406:
6404:
6399:
6398:
6395:
6387:
6383:
6382:
6377:
6373:
6372:
6368:
6362:
6360:0-486-43479-6
6356:
6352:
6351:
6345:
6341:
6335:
6331:
6326:
6323:
6322:0-03-079485-4
6319:
6315:
6314:
6309:
6305:
6302:
6299:
6297:0-356-02077-0
6293:
6289:
6284:
6281:
6280:0-387-25790-X
6277:
6273:
6269:
6266:
6265:0-13-181629-2
6262:
6258:
6254:
6251:
6248:
6247:0-07-037988-2
6244:
6240:
6236:
6232:
6227:
6223:
6222:
6217:
6212:
6209:
6208:0-387-94327-7
6205:
6201:
6197:
6194:
6191:
6187:
6184:
6179:
6177:1-4196-2722-8
6173:
6170:. Booksurge.
6169:
6168:
6163:
6162:Brown, Ronald
6159:
6156:
6152:
6149:
6146:
6145:0-387-97926-3
6142:
6138:
6134:
6131:
6127:
6125:0-387-90839-0
6121:
6117:
6113:
6108:
6107:
6103:
6094:
6090:
6086:
6082:
6078:
6074:
6071:(1): 91–119.
6070:
6066:
6065:
6057:
6053:
6049:
6043:
6040:
6035:
6031:
6027:
6023:
6019:
6015:
6011:
6007:
6004:topologies".
5989:
5985:
5962:
5958:
5946:
5943:
5938:
5932:
5928:
5921:
5918:
5914:
5909:
5906:
5902:
5897:
5894:
5890:
5885:
5882:
5878:
5873:
5870:
5866:
5864:
5851:
5849:9783110989854
5845:
5841:
5840:
5835:
5829:
5826:
5821:
5813:
5809:
5808:
5802:
5795:
5792:
5786:
5783:
5779:
5774:
5772:
5768:
5764:
5759:
5756:
5751:
5747:
5743:
5741:0-19-853155-9
5737:
5733:
5732:
5724:
5721:
5717:
5716:Schubert 1968
5712:
5709:
5702:
5697:
5694:
5688:
5685:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5660:Hilbert space
5658:
5655:
5652:
5650:
5647:
5644:
5641:
5638:
5637:Compact space
5635:
5632:
5629:
5628:
5624:
5619:
5603:
5583:
5577:
5571:
5568:
5565:
5559:
5553:
5550:
5530:
5527:
5524:
5516:
5514:
5510:
5505:
5502:
5499:
5495:
5491:
5490:
5486:: A space is
5485:
5482:
5481:
5477:
5475:
5473:
5469:
5465:
5461:
5456:
5448:
5446:
5444:
5440:
5436:
5432:
5431:connectedness
5428:
5424:
5418:
5410:
5408:
5392:
5388:
5367:
5347:
5327:
5324:
5304:
5282:
5278:
5274:
5271:
5268:
5263:
5259:
5238:
5218:
5211:
5208:
5204:
5203:Fell topology
5199:
5186:
5181:
5177:
5154:
5150:
5129:
5126:
5104:
5100:
5096:
5093:
5090:
5085:
5081:
5060:
5052:
5036:
5033:
5025:
5020:
5018:
5002:
4982:
4979:
4971:
4955:
4935:
4927:
4911:
4908:
4900:
4884:
4864:
4856:
4853:
4837:
4831:
4828:
4825:
4805:
4785:
4777:
4772:
4769:
4765:
4761:
4757:
4746:
4741:
4737:
4736:
4735:Fell topology
4731:
4730:
4725:
4720:
4711:
4710:
4704:
4702:
4689:
4684:
4680:
4657:
4653:
4645:
4641:
4637:
4624:
4621:
4601:
4581:
4558:
4555:
4552:
4529:
4523:
4520:
4517:
4495:
4489:
4486:
4483:
4472:
4453:
4450:
4447:
4441:
4438:
4430:
4414:
4405:
4368:
4362:
4359:
4356:
4345:
4340:
4338:
4333:
4331:
4324:
4319:
4317:
4313:
4308:
4306:
4302:
4297:
4295:
4291:
4287:
4283:
4278:
4276:
4272:
4271:solution sets
4256:
4251:
4222:
4207:
4203:
4199:
4194:
4192:
4188:
4184:
4180:
4175:
4173:
4168:
4166:
4162:
4159:Many sets of
4157:
4144:
4141:
4118:
4089:
4081:
4055:
4047:
4045:
4029:
4014:
3998:
3985:
3967:
3952:
3934:
3920:
3915:
3911:
3885:
3869:
3855:
3853:
3849:
3845:
3840:
3838:
3832:
3825:Metric spaces
3824:
3822:
3820:
3816:
3812:
3804:
3802:
3800:
3796:
3792:
3788:
3784:
3783:homeomorphism
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3752:
3748:
3746:
3742:
3738:
3737:homeomorphism
3734:
3733:inverse image
3718:
3698:
3695:
3692:
3686:
3680:
3660:
3640:
3617:
3611:
3591:
3571:
3568:
3565:
3558:if for every
3557:
3556:
3539:
3533:
3530:
3527:
3520:
3514:
3506:
3504:
3491:
3488:
3468:
3448:
3440:
3424:
3421:
3401:
3393:
3377:
3374:
3353:
3349:
3346:
3343:
3340:
3335:
3331:
3326:
3322:
3319:
3311:
3295:
3286:
3282:
3278:
3274:
3270:
3255:
3250:
3246:
3235:
3219:
3215:
3194:
3189:
3185:
3174:
3158:
3154:
3133:
3128:
3124:
3101:
3097:
3086:
3078:
3076:
3074:
3058:
3050:
3046:
3041:
3028:
3025:
3017:
3013:
3009:
3005:
3000:
2993:
2991:
2977:
2957:
2937:
2934:
2914:
2894:
2882:
2879:
2864:
2856:
2855:
2854:
2852:
2851:
2846:
2838:
2822:
2819:
2799:
2783:
2747:
2727:
2719:
2716:
2708:
2705:
2704:
2684:
2681:
2678:
2655:
2652:
2629:
2620:
2617:
2597:
2594:
2586:
2570:
2562:
2546:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2516:
2513:
2505:
2491:
2488:
2468:
2445:
2442:
2436:
2433:
2430:
2427:
2424:
2418:
2412:
2409:
2406:
2400:
2394:
2391:
2388:
2382:
2376:
2370:
2361:
2358:
2338:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2308:
2305:
2297:
2283:
2280:
2260:
2237:
2234:
2225:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2192:
2180:
2177:
2170:
2154:
2144:
2142:
2126:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2096:
2093:
2085:
2084:
2065:
2039:
2036:
2033:
2007:
2004:
2001:
1975:
1972:
1969:
1943:
1917:
1894:
1888:
1885:
1882:
1879:
1876:
1853:
1844:
1837:
1835:
1821:
1815:
1808:
1789:
1786:
1783:
1770:
1756:
1753:
1750:
1741:
1728:
1725:
1717:
1701:
1678:
1675:
1655:
1647:
1633:
1630:
1610:
1602:
1598:
1584:
1581:
1561:
1553:
1549:
1548:
1547:
1545:
1537:
1521:
1510:
1506:
1498:
1496:
1483:
1480:
1477:
1474:
1454:
1434:
1414:
1394:
1373:
1370:
1350:
1342:
1326:
1306:
1297:
1284:
1281:
1261:
1252:neighbourhood
1251:
1215:
1195:
1181:
1168:
1165:
1141:
1138:
1118:
1098:
1078:
1058:
1038:
1030:
1016:
1013:
993:
985:
981:
967:
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559:(or, simply,
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351:
337:
317:
309:
301:
299:
296:
293:
290:
287:
285:
281:
277:
271:
263:
261:
259:
255:
251:
247:
246:Metric spaces
243:
238:
236:
232:
228:
224:
220:
216:
212:
207:
205:
200:
196:
192:
187:
185:
181:
177:
174:(1750–1840),
173:
169:
165:
161:
145:
142:
139:
136:
133:
130:
127:
120:
116:
113:Around 1735,
108:
106:
104:
100:
95:
93:
89:
88:metric spaces
85:
81:
80:connectedness
77:
73:
69:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
6781:Publications
6646:Chern number
6636:Betti number
6535:
6519: /
6510:Key concepts
6458:Differential
6379:
6349:
6330:Set Topology
6329:
6311:
6287:
6271:
6256:
6238:
6230:
6220:
6199:
6189:
6186:ÄŚech, Eduard
6166:
6154:
6136:
6118:. Springer.
6111:
6104:Bibliography
6068:
6062:
6048:Culler, Marc
6042:
6009:
6005:
5945:
5926:
5920:
5908:
5896:
5884:
5872:
5862:
5860:
5853:. Retrieved
5838:
5828:
5805:
5794:
5785:
5758:
5730:
5723:
5711:
5512:
5508:
5503:
5487:
5483:
5472:local fields
5452:
5420:
5210:Polish space
5202:
5200:
5023:
5021:
4773:
4753:
4744:
4733:
4727:
4638:
4406:
4341:
4334:
4320:
4309:
4298:
4282:linear graph
4279:
4195:
4176:
4169:
4158:
4057:
4048:Other spaces
3950:
3884:real numbers
3856:
3852:vector space
3844:metric space
3841:
3834:
3831:Metric space
3808:
3763:
3753:
3749:homeomorphic
3553:
3516:
3287:
3088:
3042:
3008:fixed points
3001:
2997:
2886:
2848:
2842:
2701:
2700:is called a
2147:topology on
1742:
1715:
1693:
1543:
1504:
1502:
1340:
1298:
1182:
1156:
984:intersection
927:I.e., every
659:
658:is called a
606:is called a
305:
275:
273:
239:
208:
204:homeomorphic
189:Yet, "until
188:
183:
164:planar graph
112:
96:
65:
24:
18:
6744:Wikiversity
6661:Key results
6224:. Springer.
5435:compactness
4745:(June 2024)
4640:Outer space
4277:equations.
4172:local field
4011:the set of
3882:the set of
2877:are closed.
2850:closed sets
2351:the family
1668:belongs to
1623:belongs to
402:(point) in
264:Definitions
211:Felix Klein
21:mathematics
6796:Categories
6590:CW complex
6531:Continuity
6521:Closed set
6480:cohomology
6339:0486404560
6216:Xu, Dianna
6190:Point Sets
5889:Brown 2006
5877:Brown 2006
5822:required.)
5763:Gauss 1827
5049:named for
4948:for which
4852:surjective
4768:projection
4644:free group
4312:finite set
4275:polynomial
4189:and every
4052:See also:
3787:invariants
3760:categories
3673:such that
3555:continuous
2782:itself is
2145:indiscrete
1807:complement
1467:such that
1111:such that
182:published
76:continuity
6769:geometric
6764:algebraic
6615:Cobordism
6551:Hausdorff
6546:connected
6463:Geometric
6453:Continuum
6443:Algebraic
6386:EMS Press
6093:122869546
6012:: 77–81.
5855:20 August
5703:Citations
5572:
5566:⊆
5554:
5528:≤
5500:theorem).
5272:…
5094:…
4835:→
4622:γ
4602:β
4582:α
4559:γ
4553:α
4524:β
4490:β
4484:α
4454:γ
4439:γ
4415:γ
4122:Γ
4119:∪
4113:∅
4082:on a set
4066:Γ
3776:morphisms
3741:bijection
3693:⊆
3569:∈
3537:→
3390:then the
3347:∈
3344:α
3336:α
3332:τ
3247:τ
3216:τ
3186:τ
3155:τ
3125:τ
3098:τ
3016:power set
2958:τ
2915:τ
2820:τ
2748:τ
2685:τ
2624:℘
2618:τ
2585:power set
2368:∅
2359:τ
2232:∅
2178:τ
1854:τ
1819:∖
1790:τ
1754:⊆
1743:A subset
1702:τ
1676:τ
1656:τ
1631:τ
1611:τ
1582:τ
1552:empty set
1544:open sets
1542:, called
1522:τ
1478:∈
942:∈
756:∈
717:∈
289:open sets
206:or not."
172:L'Huilier
131:−
92:manifolds
61:open sets
33:closeness
31:in which
6734:Wikibook
6712:Category
6600:Manifold
6568:Homotopy
6526:Interior
6517:Open set
6475:Homology
6424:Topology
6288:Topology
6257:Topology
6218:(2013).
6164:(2006).
6054:(1986).
5927:Topology
5625:See also
5498:Hochster
5489:spectral
5484:Spectral
5453:For any
4995:The map
4855:function
4290:vertices
4179:manifold
3799:K-theory
3519:function
3437:and the
3308:forms a
3279:stronger
3049:sequence
3012:operator
1716:topology
1505:topology
929:superset
745:), then
380:function
276:topology
213:in his "
49:topology
37:distance
6759:general
6561:uniform
6541:compact
6492:Digital
6388:, 2001
6073:Bibcode
6034:0244927
6026:2037491
5750:1679102
5718:, p. 13
5251:-tuple
5073:-tuple
4740:Discuss
4187:simplex
3772:objects
3745:inverse
3275:smaller
3236:coarser
3014:on the
2583:is the
2167:is the
2141:trivial
1805:if its
1536:subsets
706:(i.e.,
610:if the
191:Riemann
119:formula
109:History
6754:Topics
6556:metric
6431:Fields
6357:
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6320:
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6245:
6206:
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4286:graphs
4208:. On
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4177:Every
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3842:Every
3837:metric
3797:, and
3770:whose
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3271:larger
3010:of an
2843:Using
2709:Given
2506:Given
2298:Given
2169:family
2086:Given
2055:[i.e.
1959:[i.e.
1771:closed
612:axioms
352:points
310:. Let
254:German
199:Jordan
195:Möbius
168:Cauchy
78:, and
72:limits
57:axioms
45:points
6536:Space
6089:S2CID
6059:(PDF)
6022:JSTOR
5816:
5423:up to
4850:is a
4724:split
4642:of a
4294:edges
4288:with
4102:then
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3785:) by
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3312:: if
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3177:than
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884:then
634:with
378:be a
6355:ISBN
6334:ISBN
6318:ISBN
6306:and
6292:ISBN
6276:ISBN
6261:ISBN
6243:ISBN
6204:ISBN
6172:ISBN
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6120:ISBN
5931:ISBN
5857:2022
5844:ISBN
5746:OCLC
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