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In mathematics, a pointed set is a set X with a distinguished basepoint 
. Maps of pointed sets (based maps) are functions preserving basepoints, i.e. a map 
such that f(x0) = y0. This is usually denoted
.
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.
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