Pointed set

In mathematics, a pointed set (also based set or rooted set) is an ordered pair ${\displaystyle (X,x_{0})}$ where ${\displaystyle X}$ is a set and ${\displaystyle x_{0}}$ is an element of ${\displaystyle X}$ called the base point (also spelled basepoint).

A map between pointed a sets ${\displaystyle (X,x_{0})}$ and ${\displaystyle (Y,y_{0})}$—called a based map, pointed map, or point-preserving map—is a function 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 simple algebraic structures. In the sense of universal algebra, a pointed set is a set ${\displaystyle X}$ together with a single nullary operation ⁠${\displaystyle *:X^{0}\to X}$⁠, which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures.

The class of all pointed sets together with the class of all based maps forms a category. Every pointed set can be converted to an ordinary set by forgetting the basepoint (the forgetful functor is faithful), but the reverse is not true. In particular, the empty set cannot be pointed, because it has no element that can be chosen as the basepoint.