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as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as
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of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a
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141:. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as
1559: – mathematical category with metric spaces as its objects and distance-non-increasing maps as its morphisms
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The reduced suspension is a functor from the category of pointed spaces to itself. This functor is
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that remains unchanged during subsequent discussion, and is kept track of during all operations.
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It is easy to see that commutativity of the diagram is equivalent to the condition that
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is usually developed on pointed spaces, and then moved to relative topologies in
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is a based map if it is continuous with respect to the topologies of
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1121:. The basepoint of the quotient is the image of the basepoint in
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which "forgets" which point is the basepoint. This functor has a
467:{\displaystyle f:\left(X,x_{0}\right)\to \left(Y,y_{0}\right).}
496:
concept is less important; it is anyway the case of a pointed
1647:
mathoverflow discussion on several base points and groupoids
1332:, which can be thought of as the 'one-point union' of spaces.
1568: – Category in mathematics where the objects are sets
702:
Such maps can be thought of as picking out a basepoint in
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Pointed spaces are often taken as a special case of the
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Gamelin, Theodore W.; Greene, Robert
Everist (1999) .
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559:. Another way to think about this category is as the
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
1002:whose single element is taken to be the basepoint.
49:. Unsourced material may be challenged and removed.
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16:Topological space with a distinguished point
383:{\displaystyle f\left(x_{0}\right)=y_{0}.}
1341:of two pointed spaces is essentially the
1326:in the category of pointed spaces is the
109:Learn how and when to remove this message
1627:Categories for the Working Mathematician
1620:
1310:{\displaystyle \left(x_{0},y_{0}\right)}
932:which assigns to each topological space
485:, where many constructions, such as the
747:{\displaystyle \{\bullet \}\downarrow }
585:{\displaystyle \{\bullet \}\downarrow }
1654:
1581:Category of topological vector spaces
1189:{\displaystyle \left(X,x_{0}\right),}
1231:{\displaystyle \left(Y,y_{0}\right)}
47:adding citations to reliable sources
18:
489:, depend on a choice of basepoint.
182:preserving basepoints, i.e., a map
13:
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695:{\displaystyle \{\bullet \}\to X.}
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1064:which shares its basepoint with
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761:for which the following diagram
477:Pointed spaces are important in
137:with a distinguished point, the
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1550: – category in mathematics
34:needs additional citations for
1572:Category of topological spaces
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627:category of topological spaces
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527:of all pointed spaces forms a
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1:
1672:Categories in category theory
1630:(second ed.). Springer.
1587:
657:{\displaystyle \{\bullet \}/}
1583: – Topological category
1057:{\displaystyle A\subseteq X}
1006:Operations on pointed spaces
995:{\displaystyle \{\bullet \}}
898:{\displaystyle \{\bullet \}}
853:{\displaystyle \{\bullet \}}
819:{\displaystyle \{\bullet \}}
614:{\displaystyle \{\bullet \}}
7:
1541:
1347:symmetric monoidal category
621:is any one point space and
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519:Category of pointed spaces
1557:Category of metric spaces
1317:serving as the basepoint.
1260:{\displaystyle X\times Y}
629:. (This is also called a
1598:Introduction to Topology
1530:{\displaystyle \Omega X}
1384:{\displaystyle \Sigma X}
1088:is basepoint preserving.
547:{\displaystyle \bullet }
390:This is usually denoted
202:between a pointed space
174:Maps of pointed spaces (
1491:taking a pointed space
1484:{\displaystyle \Omega }
1435:and the pointed circle
1415:) the smash product of
1141:under the quotient map.
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1458:{\displaystyle S^{1}.}
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43:improve this article
1602:(second ed.).
1391:of a pointed space
1355:compactly generated
1240:topological product
1097:of a pointed space
1015:of a pointed space
757:) are morphisms in
1677:Topological spaces
1624:(September 1998).
1622:Mac Lane, Saunders
1604:Dover Publications
1548:Category of groups
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718:{\displaystyle X.}
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1504:{\displaystyle X}
1428:{\displaystyle X}
1404:{\displaystyle X}
1349:with the pointed
1144:One can form the
1134:{\displaystyle X}
1110:{\displaystyle X}
1091:One can form the
1077:{\displaystyle X}
1028:{\displaystyle X}
969:{\displaystyle X}
945:{\displaystyle X}
874:forgetful functor
790:{\displaystyle f}
505:relative topology
487:fundamental group
329:{\displaystyle Y}
309:{\displaystyle X}
262:{\displaystyle Y}
215:{\displaystyle X}
195:{\displaystyle f}
135:topological space
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1566:Category of sets
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1667:Homotopy theory
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1613:0-486-40680-6
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1413:homeomorphism
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1338:smash product
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1086:inclusion map
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127:pointed space
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99:November 2009
91:
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81:
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74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1625:
1597:
1469:left adjoint
1411:is (up to a
1364:
1336:
1321:
1145:
1092:
1084:so that the
1012:
930:left adjoint
925:
876:
871:
866:
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476:
175:
173:
138:
130:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
872:There is a
828:zero object
494:pointed set
131:based space
123:mathematics
1656:Categories
1588:References
1513:loop space
1117:under any
176:based maps
69:newspapers
1522:Ω
1479:Ω
1376:Σ
1329:wedge sum
1323:coproduct
1252:×
1049:⊆
987:∙
913:→
890:∙
845:∙
811:∙
742:↓
736:∙
684:→
678:∙
644:∙
606:∙
580:↓
574:∙
557:morphisms
542:∙
433:→
139:basepoint
1662:Topology
1542:See also
1351:0-sphere
1343:quotient
1094:quotient
1013:subspace
763:commutes
633:denoted
595:) where
529:category
1511:to its
1238:as the
1147:product
625:is the
336:and if
83:scholar
1634:
1610:
178:) are
85:
78:
71:
64:
56:
1360:ones.
1267:with
1035:is a
826:is a
525:class
133:is a
90:JSTOR
76:books
1632:ISBN
1608:ISBN
1363:The
1335:The
1320:The
952:the
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