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to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts. In this case, for example, the line segment possesses infinitely many points, and therefore cannot be put into a bijection with a set containing only a finite number of
170:
are not. However, this description can be misleading. Some continuous deformations do not result into homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from continuous deformations, such as the homeomorphism between a
1542:
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deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces:
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The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly—it may not be obvious from the description above that deforming a
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to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space
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by creating a dimple and progressively enlarging it, while preserving the donut hole in the mug's handle. This illustrates that a coffee mug and a donut (
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1425:
are homeomorphic; since the unit disc can be deformed into the unit square. An example of a bicontinuous mapping from the square to the disc is, in
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There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an
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Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them,
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1537:{\displaystyle (\rho ,\theta )\mapsto \left({\tfrac {\rho }{\max(|\cos \theta |,|\sin \theta |)}},\theta \right).}
874:. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms
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In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other.
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object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a
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Differential
Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems
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This characterization of a homeomorphism often leads to a confusion with the concept of
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are equivalence relations that have been introduced for dealing with such situations.
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is a homeomorphism from a topological space onto itself. Being "homeomorphic" is an
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44:"Topological equivalence" redirects here. For the concept in dynamical systems, see
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will coincide. Note however that this does not extend to properties defined via a
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while other such mappings are given by scaled and translated versions of the
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but the points it maps to numbers in between lie outside the neighbourhood.
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is a homeomorphism between the domain of the parametrization and the curve.
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of a given space. Two spaces with a homeomorphism between them are called
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2387:. Texts in Applied Mathematics. Vol. 18. Springer. p. 204.
2308: – Uniformly continuous homeomorphism is an isomorphism between
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2352: – Group of isotopy classes of a topological automorphism group
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This function is bijective and continuous, but not a homeomorphism (
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are not homeomorphic because one is compact while the other is not.
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of this point also includes points that the function maps close to
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27:
Mapping which preserves all topological properties of a given space
2445:. Translated by Stillwell, John. American Mathematical Society.
2346: – Continuous deformation between two continuous functions
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Continuous mappings are not always realizable as deformations.
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Papers on
Topology: Analysis Situs and Its Five Supplements
270:
2183:
can be extended to a self-homeomorphism of the whole disk
1276:(In this case, a bicontinuous forward mapping is given by
1628:
with a single point removed and the set of all points in
59:
is that topologists cannot tell the difference between a
2296: – Mathematical function revertible near each point
1955:
is not homeomorphic to the unit circle as a subspace of
2520:"On Homeomorphism Groups and the Compact-Open Topology"
2319:
Pages displaying short descriptions of redirect targets
2317: – Distance-preserving mathematical transformation
610:{\textstyle f(\varphi )=(\cos \varphi ,\sin \varphi ).}
2417:. Journal de l'Ecole polytechnique. Gauthier-Villars.
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are precise definitions for the informal concept of
1340:{\textstyle f(x)={\frac {1}{a-x}}+{\frac {1}{b-x}}}
464:, is essential. Consider for instance the function
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948:This group can be given a topology, such as the
150:Very roughly speaking, a topological space is a
2302: – Isomorphism of differentiable manifolds
2114:, then the other is as well; if one of them is
2110:, then the other is as well; if one of them is
1595:is a homeomorphism between the unit sphere in
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952:, which under certain assumptions makes it a
1079:and, given a specific homeomorphism between
2381:Hubbard, John H.; West, Beverly H. (1995).
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2240:as a continuous deformation, but from one
1138:is homeomorphic to a solid torus, but not
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30:For homeomorphisms in graph theory, see
2556:from the original on 16 September 2016.
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2102:Two homeomorphic spaces share the same
14:
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2466:Gamelin, T. W.; Greene, R. E. (1999).
2218:
363:A homeomorphism is sometimes called a
162:are homeomorphic to each other, but a
2592:
2358: – Theorem in geometric topology
2137:A homeomorphism is simultaneously an
367:function. If such a function exists,
2518:Dijkstra, Jan J. (1 December 2005).
238:if it has the following properties:
2472:(2nd ed.). Dover. p. 67.
2118:, then the other is as well; their
1867:
71:could be reshaped to the form of a
24:
2229:points, including a single point.
1728:is a homeomorphism. Also, for any
99:meaning "similar shape", named by
25:
3019:
2564:
2527:The American Mathematical Monthly
2106:. For example, if one of them is
2025:but the real line is not compact.
1856:{\displaystyle y\mapsto xyx^{-1}}
1169:{\displaystyle \mathbb {R} ^{3}.}
1003:{\textstyle {\text{Homeo}}(X,Y),}
511:{\textstyle f:[0,2\pi )\to S^{1}}
2971:
2944:
2934:
2924:
2913:
2903:
2902:
2696:
2252:—one just follows them as
2016:{\displaystyle \mathbb {R} ^{2}}
1979:{\displaystyle \mathbb {R} ^{2}}
1929:{\displaystyle \mathbb {R} ^{n}}
1896:{\displaystyle \mathbb {R} ^{m}}
1652:{\displaystyle \mathbb {R} ^{2}}
1619:{\displaystyle \mathbb {R} ^{3}}
1416:{\displaystyle \mathbb {R} ^{2}}
546:{\displaystyle \mathbb {R} ^{2}}
1721:{\displaystyle x\mapsto x^{-1}}
1122:all three sets are identified.
1072:{\textstyle {\text{Homeo}}(Y),}
941:{\textstyle {\text{Homeo}}(X).}
713:is not continuous at the point
67:, since a sufficiently pliable
2998:Theory of continuous functions
2511:
2486:
2459:
2401:
2374:
2248:correspond to which points on
2079:
2067:
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2035:
2028:The one-dimensional intervals
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1580:is a homeomorphism between an
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1038:{\textstyle {\text{Homeo}}(X)}
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133:category of topological spaces
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1:
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1014:for the homeomorphism groups
190:
135:—that is, they are the
2618:
2338:Homeomorphism (graph theory)
2156:Every self-homeomorphism in
1814:{\displaystyle y\mapsto yx,}
1782:{\displaystyle y\mapsto xy,}
1238:{\displaystyle \mathbb {R} }
430:The third requirement, that
32:Homeomorphism (graph theory)
7:
2577:Encyclopedia of Mathematics
2287:
2276:and the homeomorphism from
1992:as a subspace of Euclidean
1988:, since the unit circle is
1821:and the inner automorphism
1125:
419:on topological spaces. Its
10:
3024:
2865:Banach fixed-point theorem
2321:is an isomorphism between
43:
36:
29:
2898:
2855:
2819:
2705:
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2626:
1938:are not homeomorphic for
127:. Homeomorphisms are the
87:and more specifically in
2497:. Limes RY. p. 63.
2469:Introduction to Topology
2441:Poincaré, Henri (2010).
1593:stereographic projection
223:{\displaystyle f:X\to Y}
37:Not to be confused with
2493:Väisälä, Jussi (1999).
2362:Universal homeomorphism
1750:{\displaystyle x\in G,}
1554:is homeomorphic to the
1552:differentiable function
1218:is homeomorphic to the
866:Homeomorphisms are the
105:topological isomorphism
3003:Functions and mappings
2920:Mathematics portal
2820:Metrics and properties
2806:Second-countable space
2204:
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2104:topological properties
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1789:the right translation
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683:is not). The function
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676:{\textstyle [0,2\pi )}
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329:
328:{\displaystyle f^{-1}}
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185:continuous deformation
141:topological properties
139:that preserve all the
123:that has a continuous
80:
2315:Isometric isomorphism
2205:
2203:{\displaystyle D^{2}}
2178:
2176:{\displaystyle S^{1}}
2134:and the other is not.
2087:
2085:{\displaystyle (0,1)}
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2018:
1981:
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1757:the left translation
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950:compact-open topology
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109:bicontinuous function
54:
46:Topological conjugacy
2875:Invariance of domain
2827:Euler characteristic
2801:Bundle (mathematics)
2259:homotopy equivalence
2236:, which is actually
2187:
2160:
2064:
2032:
1998:
1961:
1911:
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1269:{\textstyle a<b.}
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2885:Tychonoff's theorem
2880:Poincaré conjecture
2634:General (point-set)
2356:Poincaré conjecture
2350:Mapping class group
2328:Homeomorphism group
2306:Uniform isomorphism
2294:Local homeomorphism
2219:Informal discussion
2145:; that is, it maps
1863:are homeomorphisms.
907:homeomorphism group
893:{\textstyle X\to X}
856:{\textstyle 2\pi ,}
771:{\textstyle f^{-1}}
741:{\textstyle (1,0),}
706:{\textstyle f^{-1}}
453:{\textstyle f^{-1}}
421:equivalence classes
117:continuous function
79:) are homeomorphic.
2870:De Rham cohomology
2791:Polyhedral complex
2781:Simplicial complex
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1381:{\textstyle D^{2}}
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232:topological spaces
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121:topological spaces
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55:An often-repeated
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2774:fundamental group
2479:978-0-486-40680-0
2452:978-0-8218-5234-7
2394:978-0-387-94377-0
2212:Alexander's trick
2149:to open sets and
1690:topological group
1681:{\displaystyle G}
1661:(a 2-dimensional
1561:A differentiable
1517:
1427:polar coordinates
1335:
1314:
1092:{\displaystyle X}
1055:
1024:
980:
954:topological group
924:
748:because although
400:{\displaystyle Y}
380:{\displaystyle X}
348:{\displaystyle f}
288:{\displaystyle f}
254:{\displaystyle f}
57:mathematical joke
16:(Redirected from
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2425:. Archived from
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2344:Homotopy#Isotopy
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2053:{\displaystyle }
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97:from Greek roots
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2894:
2890:Urysohn's lemma
2851:
2815:
2701:
2692:
2664:low-dimensional
2622:
2617:
2572:"Homeomorphism"
2570:
2567:
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2553:
2533:(10): 910–912.
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2429:on 11 June 2016
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1586:Euclidean space
1563:parametrization
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826:{\textstyle 0,}
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335:is continuous (
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103:), also called
49:
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5:
3021:
3011:
3010:
3008:Homeomorphisms
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3000:
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2837:Winding number
2834:
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2769:homotopy group
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2566:
2565:External links
2563:
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2510:
2503:
2485:
2478:
2458:
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2414:Analysis Situs
2400:
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2310:uniform spaces
2303:
2300:Diffeomorphism
2297:
2289:
2286:
2220:
2217:
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2197:
2193:
2170:
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2154:
2143:closed mapping
2135:
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2005:
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1951:The Euclidean
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175:and a circle.
101:Henri Poincaré
26:
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2848:
2847:Orientability
2845:
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2720:
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2684:
2683:Set-theoretic
2681:
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2672:
2669:
2665:
2662:
2661:
2660:
2657:
2655:
2652:
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2647:
2645:
2644:Combinatorial
2642:
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2504:951-745-184-9
2500:
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2323:metric spaces
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93:homeomorphism
90:
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74:
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66:
62:
58:
53:
47:
40:
33:
19:
2977:Publications
2842:Chern number
2832:Betti number
2715: /
2706:Key concepts
2654:Differential
2575:
2530:
2526:
2513:
2494:
2488:
2468:
2461:
2442:
2431:. Retrieved
2427:the original
2413:
2409:Poincaré, H.
2403:
2383:
2376:
2281:
2277:
2273:
2270:identity map
2268:between the
2263:
2253:
2249:
2245:
2241:
2237:
2231:
2226:line segment
2222:
2139:open mapping
1944:
1940:
1220:real numbers
1136:trefoil knot
1134:A thickened
969:
958:
911:
905:
868:isomorphisms
865:
555:) defined by
429:
424:
412:
409:homeomorphic
408:
365:bicontinuous
364:
362:
357:open mapping
235:
230:between two
194:
184:
173:trefoil knot
149:
145:homeomorphic
144:
129:isomorphisms
108:
104:
92:
82:
39:homomorphism
18:Homeomorphic
2940:Wikiversity
2857:Key results
2495:Topologia I
2151:closed sets
1582:open subset
1390:unit square
1358:The unit 2-
1355:functions).
520:unit circle
423:are called
85:mathematics
2992:Categories
2786:CW complex
2727:Continuity
2717:Closed set
2676:cohomology
2368:References
2333:Dehn twist
2097:Properties
462:continuous
297:continuous
267:one-to-one
191:Definition
73:coffee mug
61:coffee mug
2965:geometric
2960:algebraic
2811:Cobordism
2747:Hausdorff
2742:connected
2659:Geometric
2649:Continuum
2639:Algebraic
2582:EMS Press
2423:715734142
2147:open sets
2116:Hausdorff
2112:connected
1953:real line
1846:−
1832:↦
1800:↦
1768:↦
1739:∈
1711:−
1703:↦
1524:θ
1506:θ
1503:
1484:θ
1481:
1463:ρ
1452:↦
1446:θ
1440:ρ
1329:−
1308:−
1183:The open
885:→
848:π
761:−
696:−
668:π
599:φ
596:
587:φ
584:
569:φ
496:→
490:π
443:−
318:−
263:bijection
215:→
152:geometric
113:bijective
2930:Wikibook
2908:Category
2796:Manifold
2764:Homotopy
2722:Interior
2713:Open set
2671:Homology
2620:Topology
2551:Archived
2547:30037630
2433:29 April
2411:(1895).
2288:See also
2242:function
2234:homotopy
2132:complete
2120:homotopy
1943:≠
1578:manifold
1388:and the
1353:arg tanh
1247:for any
1185:interval
1140:isotopic
1126:Examples
961:Homotopy
900:forms a
197:function
177:Homotopy
137:mappings
119:between
89:topology
2955:general
2757:uniform
2737:compact
2688:Digital
2584:, 2001
2266:isotopy
2238:defined
2108:compact
2023:
1994:
1990:compact
1986:
1957:
1936:
1907:
1903:
1874:
1659:
1630:
1626:
1597:
1423:
1394:
1245:
1223:
1176:
1144:
965:isotopy
870:in the
646:compact
553:
524:
181:isotopy
131:in the
111:, is a
2950:Topics
2752:metric
2627:Fields
2545:
2501:
2476:
2449:
2421:
2391:
2141:and a
2128:metric
1556:domain
1012:torsor
355:is an
166:and a
164:sphere
160:circle
158:and a
156:square
63:and a
2732:Space
2554:(PDF)
2543:JSTOR
2523:(PDF)
1688:is a
1663:plane
1576:of a
1574:chart
1567:curve
1565:of a
1550:of a
1548:graph
1054:Homeo
1023:Homeo
1010:is a
979:Homeo
963:and
923:Homeo
902:group
778:maps
518:(the
261:is a
234:is a
168:torus
107:, or
77:torus
69:donut
65:donut
2499:ISBN
2474:ISBN
2447:ISBN
2435:2018
2419:OCLC
2389:ISBN
2122:and
2060:and
1905:and
1591:The
1546:The
1360:disc
1258:<
1142:in
1099:and
1045:and
833:any
648:but
411:. A
407:are
387:and
302:the
271:onto
269:and
179:and
115:and
91:, a
2535:doi
2531:112
2280:to
2272:on
1668:If
1500:sin
1478:cos
1467:max
1392:in
1351:or
1349:tan
910:of
810:to
644:is
593:sin
581:cos
522:in
460:be
295:is
83:In
2994::
2580:,
2574:,
2549:.
2541:.
2529:.
2525:.
2284:.
2261:.
2214:).
1665:).
1572:A
1429:,
956:.
427:.
359:).
273:),
195:A
187:.
2612:e
2605:t
2598:v
2537::
2507:.
2482:.
2455:.
2437:.
2397:.
2282:Y
2278:X
2274:X
2254:X
2250:Y
2246:X
2210:(
2196:2
2192:D
2169:1
2165:S
2080:)
2077:1
2074:,
2071:0
2068:(
2048:]
2045:1
2042:,
2039:0
2036:[
2009:2
2004:R
1972:2
1967:R
1947:.
1945:n
1941:m
1922:n
1917:R
1889:m
1884:R
1849:1
1842:x
1838:y
1835:x
1829:y
1809:,
1806:x
1803:y
1797:y
1777:,
1774:y
1771:x
1765:y
1745:,
1742:G
1736:x
1714:1
1707:x
1700:x
1676:G
1645:2
1640:R
1612:3
1607:R
1588:.
1532:.
1528:)
1521:,
1514:)
1510:|
1496:|
1492:,
1488:|
1474:|
1470:(
1456:(
1449:)
1443:,
1437:(
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1404:R
1374:2
1370:D
1332:x
1326:b
1322:1
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1311:x
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1301:1
1296:=
1293:)
1290:x
1287:(
1284:f
1264:.
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