Inmathematics, apointed set^[1][2] (alsobased set^[1] orrooted set^[3]) is anordered pair${\displaystyle (X,x_0)}$ where${\displaystyle X}$ is aset and${\displaystyle x_0}$ is an element of${\displaystyle X}$ called thebase point^[2] (also spelledbasepoint).^{[4]: 10–11}

Amap between pointed a sets${\displaystyle (X,x_0)}$ and${\displaystyle (Y,y_0)}$—called abased map,^[5]pointed map,^[4] orpoint-preserving map^[6]—is afunction from${\displaystyle X}$ to${\displaystyle Y}$ that maps one basepoint to another, i.e. a map${\displaystyle f:X\to Y}$ such that⁠${\displaystyle f(x_0)=y_0}$⁠. A based map is usually denoted⁠${\displaystyle f:(X,x_0)\to (Y,y_0)}$⁠.

Pointed sets are very simplealgebraic structures. In the sense ofuniversal algebra, a pointed set is a set${\displaystyle X}$ together with a singlenullary operation⁠${\displaystyle \ast :X^0\to X}$⁠,^[a] which picks out the basepoint.^[7] Pointed maps are thehomomorphisms of these algebraic structures.

Theclass of all pointed sets together with the class of all based maps forms acategory. Every pointed set can be converted to an ordinary set by forgetting the basepoint (theforgetful functor isfaithful), but the reverse is not true.^[8]: 44 In particular, theempty set cannot be pointed, because it has no element that can be chosen as the basepoint.^[9]

The category of pointed sets and based maps is equivalent to the category of sets andpartial functions.^[6] The base point serves as a "default value" for those arguments for which the partial function is not defined. One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, intopology (one-point compactification) and intheoretical computer science."^[10] This category is also isomorphic to thecoslice category (⁠${\displaystyle \mathbf{1}\downarrow \mathbf{S}\mathbf{e}\mathbf{t}}$⁠), where${\displaystyle \mathbf{1}}$ is (a functor that selects) a singleton set, and${\displaystyle \mathbf{S}\mathbf{e}\mathbf{t}}$ (the identity functor of) thecategory of sets.^{[8]: 46 [11]} This coincides with the algebraic characterization, since the unique map${\displaystyle \mathbf{1}\to \mathbf{1}}$ extends thecommutative triangles defining arrows of the coslice category to form thecommutative squares defining homomorphisms of the algebras.

There is afaithful functor from pointed sets to usual sets, but it is not full and these categories are notequivalent.^[8]

The category of pointed sets is apointed category. The pointedsingleton sets${\displaystyle (\{a\},a)}$ are bothinitial objects andterminal objects,^[1] i.e. they arezero objects.^[4]: 226 The category of pointed sets and pointed maps has bothproducts andcoproducts, but it is not adistributive category. It is also an example of a category where${\displaystyle 0\times A}$ is not isomorphic to⁠${\displaystyle 0}$⁠.^[9]

Manyalgebraic structures rely on a distinguished point. For example,groups are pointed sets by choosing theidentity element as the basepoint, so thatgroup homomorphisms are point-preserving maps.^[12]: 24 This observation can be restated in category theoretic terms as the existence of aforgetful functor from groups to pointed sets.^[12]: 582

A pointed set may be seen as apointed space under thediscrete topology or as avector space over thefield with one element.^[13]

As "rooted set" the notion naturally appears in the study ofantimatroids^[3] and transportation polytopes.^[14]

• Accessible pointed graph
• Alexandroff extension – Way to extend a non-compact topological space
• Riemann sphere – Model of the extended complex plane plus a point at infinity

• Lawvere, F. W.; Schanuel, Stephen Hoel (2009).Conceptual Mathematics: A First Introduction to Categories (2nd ed.). Cambridge University Press. pp. 296–298.ISBN 978-0-521-89485-2.
• Mac Lane, Saunders (1998).Categories for the Working Mathematician (2nd ed.). Springer-Verlag.ISBN 0-387-98403-8.Zbl 0906.18001.
• Schröder, Lutz (2001). "Categories: a free tour". In Koslowski, Jürgen; Melton, Austin (eds.).Categorical Perspectives. Springer Science & Business Media. p. 10.ISBN 978-0-8176-4186-3.

• Pullbacks in Category of Sets and Partial Functions
• Pointed set atPlanetMath.
• Pointed object at thenLab

• Basic concepts in set theory
• Algebraic structures
• Category theory

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