Pointed set
From Citizendium, the Citizens' Compendium
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 
where 
. A mapping from the pointed set to 
is a function 
such that 
.
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.
