Pointed set

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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.