Editing Pointed set (section)

==Applications==
Many [[variety (universal algebra)|algebraic structures]] rely on a distinguished point. For example, [[Group (mathematics)|groups]] are pointed sets by choosing the [[identity element]] as the basepoint, so that [[group homomorphism]]s are point-preserving maps.<ref name="Aluffi2009">{{cite book|author=Paolo Aluffi|title=Algebra: Chapter 0|year=2009|publisher=American Mathematical Soc.|isbn=978-0-8218-4781-7}}</ref>{{rp|24}} This observation can be restated in category theoretic terms as the existence of a [[forgetful functor]] from groups to pointed sets.<ref name="Aluffi2009"/>{{rp|582}}

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]].<ref>{{citation |last=Haran |first=M. J. Shai |title=Non-additive geometry |url=http://cage.ugent.be/~kthas/Fun/library/ShaiHaran2007.pdf |journal=Compositio Mathematica |volume=143 |issue=3 |pages=618–688 |year=2007 |doi=10.1112/S0010437X06002624 |doi-broken-date=8 February 2025 |mr=2330442 |author-link=Shai Haran}}. On p.&nbsp;622, Haran writes "We consider <math>\mathbb{F}</math>-vector spaces as finite sets <math>X</math> with a distinguished 'zero' element..."</ref>

As "rooted set" the notion naturally appears in the study of [[antimatroid]]s<ref name="Greedoids">{{citation | last1=Korte | first1=Bernhard |author-link1=Bernhard Korte | last2=Lovász | first2=László | author-link2=László Lovász | last3=Schrader | first3=Rainer | year=1991 | title=Greedoids | location=New York, Berlin | publisher=[[Springer-Verlag]] | series=Algorithms and Combinatorics | volume=4 | isbn=3-540-18190-3 | zbl=0733.05023 | at=chapter 3}}</ref> and transportation polytopes.<ref>{{cite book|editor=George Bernard Dantzig|title=Mathematics of the Decision Sciences. Part 1|year=1970|orig-year=1968|publisher=American Mathematical Soc.|chapter= Facets and vertices of transportation polytopes|first1=V. | last1=Klee | first2=C. | last2=Witzgall|oclc=859802521|asin=B0020145L2}}</ref>