# Pointed set

In mathematics, a pointed set is a set $X$ with a distinguished element $x_0\in X$, which is called the basepoint. Maps of pointed sets (based maps) are those functions that map one basepoint to another, i.e. a map $f : X \to Y$ such that $f(x_0) = y_0$. This is usually denoted

$f : (X, x_0) \to (Y, y_0)$.

Pointed sets may be regarded as a rather simple algebraic structure. In the sense of universal algebra, they are structures with a single nullary operation which picks out the basepoint.

The class of all pointed sets together with the class of all based maps form a category.

A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the field with one element.

There is a faithful functor from usual sets to pointed sets, but it is not full, and these categories are not equivalent.