Pointed set

In mathematics, a pointed set^[1][2] (also based set^[1] or rooted set^[3]) 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 ^[2] (also spelled basepoint).^{[4]: 10–11}

Maps between pointed sets ${\displaystyle (X,x_{0})}$ and ${\displaystyle (Y,y_{0})}$—called based maps,^[5] pointed maps,^[4] or point-preserving maps^[6]—are functions from ${\displaystyle X}$ to ${\displaystyle Y}$ that map one basepoint to another, i.e. maps ${\displaystyle f\colon X\to Y}$ such that ${\displaystyle f(x_{0})=y_{0}}$. Based maps are usually denoted ${\textstyle f\colon (X,x_{0})\to (Y,y_{0})}$.

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 ${\displaystyle *:X^{0}\to X,}$^[a] which picks out the basepoint.^[7] 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.^[8]: 44 In particular, the empty set cannot be pointed, because it has no element that can be chosen as the basepoint.^[9]

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