Pointed set

In set theory, a pointed set is a set together with a distinguished element, known as the base point. Mappings between pointed sets are assumed to respect the base point.

Formally, a pointed set is a pair $(X,b)$ where $b\in X$. A mapping from the pointed set $(X,b)$ to $(Y,c)$ is a function $f:X\rightarrow Y$ such that $f(b)=c$.

Examples

• Many algebraic structures such as a monoid, group or vector space have a distinguished element, such as an identity element, and morphisms of the structures respect those elements.
• In homotopy theory, the fundamental group of a topological space is defined in terms of a base point.
• Choice of base point is the distinction between certain types of structure:
  • Principal homogeneous space versus abelian group;
  • Affine space versus vector space;
  • Algebraic curve of genus one versus elliptic curve.