Thomson's lamp: Difference between revisions

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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>.
 
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.
 
The [[Class (set theory)|class]] of all pointed sets together with the class of all based maps form a [[category theory|category]].
 
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]].
 
There is a faithful functor from usual sets to pointed sets, but it is not full, and these categories are not equivalent.
 
== References ==
* {{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 }}
* {{cite book
|last=Mac Lane
|first=Saunders
|authorlink=Saunders Mac Lane
|title=[[Categories for the Working Mathematician]]
|publisher=Springer-Verlag
|year=1998
|edition=2nd
|isbn=0-387-98403-8
}}
 
{{DEFAULTSORT:Pointed Set}}
[[Category:Basic concepts in set theory]]
[[Category:Abstract algebra]]

Latest revision as of 06:29, 6 January 2015

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