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| In [[mathematics]], a '''pointed set''' is a [[Set (mathematics)|set]] <math>X</math> with a distinguished element <math>x_0\in X</math>, which is called the '''basepoint'''. Maps of pointed sets ('''based maps''') are those [[function (mathematics)|functions]] that map one basepoint to another, i.e. a map <math>f : X \to Y</math> such that <math>f(x_0) = y_0</math>. This is usually denoted
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| :<math>f : (X, x_0) \to (Y, y_0)</math>. | |
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| Pointed sets may be regarded as a rather simple [[algebraic structure]]. In the sense of [[universal algebra]], they are structures with a single [[nullary operation]] which picks out the basepoint.
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| The [[Class (set theory)|class]] of all pointed sets together with the class of all based maps form a [[category theory|category]].
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| A pointed set may be seen as a [[pointed space]] under the [[discrete topology]] or as a [[vector space]] over the [[field with one element]].
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| There is a faithful functor from usual sets to pointed sets, but it is not full, and these categories are not equivalent.
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| == References ==
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| * {{cite book | title=An Introduction to Galois Cohomology and Its Applications | volume=377 | series=London Mathematical Society Lecture Note Series | author=Grégory Berhuy | publisher=Cambridge University Press | year=2010 | isbn=0-521-73866-0 | page=34 }}
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| * {{cite book
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| |last=Mac Lane
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| |first=Saunders
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| |authorlink=Saunders Mac Lane
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| |title=[[Categories for the Working Mathematician]]
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| |publisher=Springer-Verlag
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| |year=1998
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| |edition=2nd
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| |isbn=0-387-98403-8
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| }}
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| {{DEFAULTSORT:Pointed Set}}
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| [[Category:Basic concepts in set theory]]
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| [[Category:Abstract algebra]]
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Latest revision as of 06:29, 6 January 2015
I'm Luther but I never really liked that title. One of the things I adore most is climbing and now I have time to take on new issues. My job is a messenger. Her husband and her selected to reside in Alabama.
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