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.

References

• Grégory Berhuy (2010). An Introduction to Galois Cohomology and Its Applications. London Mathematical Society Lecture Note Series. Vol. 377. Cambridge University Press. p. 34. ISBN 0521738660.