Integer and digital spaces are playing a significant role in digital image
processing, computer graphics, computer tomography, robot vision,
and many other fields dealing with finitely or countable many objects.
It is proven here that every finite T0-space is a quotient space of a
subspace of some simplex, i.e. of some subspace of a Euclidean space.
Thus finite and digital spaces can be considered as abstract simplicial
structures of subspaces of Euclidean spaces. Primitive subspaces of
finite, digital, and integer spaces are introduced. They prove to be
useful in the investigation of connectedness structure, which can be
represented as a poset, and also in consideration of the dimension of
finite spaces. Essentially T0-spaces and finitely connected and
primitively path connected spaces are discussed.
