In The Elements, Euclid defines a point as that which has no part.

In a vector space, an affine space, or, more generally, an incidence geometry, a point is a zero dimensional object.

In a projective geometry, a point is a one-dimensional subspace of the vector space underlying the projective geometry.

In a topology, a point is an element of a topological space.

Note that there is also the possibility for a point-free approach to geometry in which points are not assumed as a primitive. Instead, points are defined by suitable abstraction processes. (See point-free geometry.)