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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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152:. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as
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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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1132:. 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
478:{\displaystyle f:\left(X,x_{0}\right)\to \left(Y,y_{0}\right).}
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concept is less important; it is anyway the case of a pointed
1658:
mathoverflow discussion on several base points and groupoids
1343:, which can be thought of as the 'one-point union' of spaces.
1579: – Category in mathematics where the objects are sets
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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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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
1013:whose single element is taken to be the basepoint.
60:. Unsourced material may be challenged and removed.
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27:Topological space with a distinguished point
394:{\displaystyle f\left(x_{0}\right)=y_{0}.}
1352:of two pointed spaces is essentially the
1337:in the category of pointed spaces is the
120:Learn how and when to remove this message
1638:Categories for the Working Mathematician
1631:
1321:{\displaystyle \left(x_{0},y_{0}\right)}
943:which assigns to each topological space
496:, where many constructions, such as the
758:{\displaystyle \{\bullet \}\downarrow }
596:{\displaystyle \{\bullet \}\downarrow }
14:
1665:
1592:Category of topological vector spaces
1200:{\displaystyle \left(X,x_{0}\right),}
1242:{\displaystyle \left(Y,y_{0}\right)}
58:adding citations to reliable sources
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500:, depend on a choice of basepoint.
193:preserving basepoints, i.e., a map
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706:{\displaystyle \{\bullet \}\to X.}
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1583:Category of topological spaces
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638:category of topological spaces
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538:of all pointed spaces forms a
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1683:Categories in category theory
1641:(second ed.). Springer.
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668:{\displaystyle \{\bullet \}/}
1594: – Topological category
1068:{\displaystyle A\subseteq X}
1017:Operations on pointed spaces
1006:{\displaystyle \{\bullet \}}
909:{\displaystyle \{\bullet \}}
864:{\displaystyle \{\bullet \}}
830:{\displaystyle \{\bullet \}}
625:{\displaystyle \{\bullet \}}
7:
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1358:symmetric monoidal category
632:is any one point space and
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530:Category of pointed spaces
1568:Category of metric spaces
1328:serving as the basepoint.
1271:{\displaystyle X\times Y}
640:. (This is also called a
18:Pointed topological space
1609:Introduction to Topology
1541:{\displaystyle \Omega X}
1395:{\displaystyle \Sigma X}
1099:is basepoint preserving.
558:{\displaystyle \bullet }
401:This is usually denoted
213:between a pointed space
185:Maps of pointed spaces (
1502:taking a pointed space
1495:{\displaystyle \Omega }
1446:and the pointed circle
1426:) the smash product of
1152:under the quotient map.
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1469:{\displaystyle S^{1}.}
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54:improve this article
1613:(second ed.).
1402:of a pointed space
1366:compactly generated
1251:topological product
1108:of a pointed space
1026:of a pointed space
768:) are morphisms in
1688:Topological spaces
1635:(September 1998).
1633:Mac Lane, Saunders
1615:Dover Publications
1559:Category of groups
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1515:{\displaystyle X}
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1360:with the pointed
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1121:{\displaystyle X}
1102:One can form the
1088:{\displaystyle X}
1039:{\displaystyle X}
980:{\displaystyle X}
956:{\displaystyle X}
885:forgetful functor
801:{\displaystyle f}
516:relative topology
498:fundamental group
340:{\displaystyle Y}
320:{\displaystyle X}
273:{\displaystyle Y}
226:{\displaystyle X}
206:{\displaystyle f}
146:topological space
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16:(Redirected from
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1424:homeomorphism
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1349:smash product
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1097:inclusion map
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110:November 2009
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71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
1636:
1608:
1480:left adjoint
1422:is (up to a
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1103:
1095:so that the
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941:left adjoint
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90:
83:
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64:
52:Please help
47:verification
44:
883:There is a
839:zero object
505:pointed set
142:based space
134:mathematics
1667:Categories
1599:References
1524:loop space
1128:under any
187:based maps
80:newspapers
1533:Ω
1490:Ω
1387:Σ
1340:wedge sum
1334:coproduct
1263:×
1060:⊆
998:∙
924:→
901:∙
856:∙
822:∙
753:↓
747:∙
695:→
689:∙
655:∙
617:∙
591:↓
585:∙
568:morphisms
553:∙
444:→
150:basepoint
1673:Topology
1553:See also
1362:0-sphere
1354:quotient
1105:quotient
1024:subspace
774:commutes
644:denoted
606:) where
540:category
1522:to its
1249:as the
1158:product
636:is the
347:and if
94:scholar
1645:
1621:
189:) are
96:
89:
82:
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1371:ones.
1278:with
1046:is a
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