Pointed set

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In mathematics, a pointed set is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the basepoint.[1]

Maps of between pointed sets (X, x_0) and (Y, y_0) (based maps) are functions from X to Y 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.

See also[edit]


  1. ^ Pointed Set at PlanetMath