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;
 * Algebraic curve of genus one versus elliptic curve.