Pointed set

Pointed set

In mathematics, a pointed set is a set "X" with a distinguished basepoint "x"0 in "X". Maps of pointed sets (based maps) are functions preserving basepoints, i.e. a map "f" : "X" → "Y" such that "f"("x"0) = "y"0. This is usually denoted:"f" : ("X", "x"0) → ("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.


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