Some notes on a most general definition of "geometry",
first illucidated by Felix Klein that is based on
a set of geometric invariants under a group of tramsformations.
Klein's "Erlanger Program".

According to Felix Klein: A space S is coordinatized by (x_1, x_2, ..., x_n), i.e., there is a bijective map between the points of S and the coordinates. The coordinates are usually local functions on S with values in a field. Relationships between the coordinates of points can be specified. There is an abstract group g acting on S which has a representation as a group of linear substitutions on the coordintes. (could be homogeneous or inhomogeneous)

A "geometry" is defined by the set of relations among the coordinates that are invariant under the group of coordinate substitutions.

Thus, generally, a geometry is defined by a space (set) S, a set R of relations, and the group g of transformations of S that leave R invariant, since the set R is implied by the action of g on S, Thus, most generally the coordinates can be conceptually eliminated.

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Ultimately an abstract geometry G is defined simply by a set S
and a group g which acts as a transformation group on S.
The relations on S that are left invariant by g are called G-geometric.
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Example: S is an R^n and g is an R^n which is
an abelian group, (a group of translations)
(S, g) is then a geometry with translationally invariant
relations which may not be rotationally invariant.

Example: S is a Euclidean space and g is IO(n) which is
the semidirect product of the full group of rotations O(n)
and an abelian group, (a group of translations) isomorphic
to R^n.
(S, g) is a homogeneous space.

Example: Homogeneous spaces
Let S be a space upon which a Lie Group g acts effectively.
Let h be an isotropy subgroup of g in its action on S, then

S is isomorphic to g/h

This generalizes the second example.

Underlying Klein's notion of geometry are the following modifiable
assumptions:

1) S will be LOCALLY homeomorphic to an R^n.
That is, S will be a classical manifold.
[Essential source of coordinates as n-tuples of field elements]
[Essential source of dimension as a topological invariant;
possibility of recursive definition of Brouwer-Uryhson]

2) Structurally, the R^n is an Abelian ring over (the real)
field, hence actually a commutative algebra,
though only the vector space properties are necessary.
Point-point multiplication is not a part of classical
geometry and has no meaning in classical intuitive geometry.
[Essential source of intrinsic geometric holonomy]

3) The field (of reals) has characteristic zero.
[Essential source of continuousness of the manifold]
Note that finite fields have an implicit toroidal
topology, and that there is a concept of nearness
which can be refined by a definition of a topology
that is a proper subset of the power set.
Alternatively, develop "proximity" independently
of a topological notion, perhaps more along the line
a sigma algebra that supports a measure on the Galois
field.

See also, a review of the historical concepts of Euclidean geometry upon which a modern algebraic understanding of more general geometry is based. Classical Geometry & Physics Redux, which is logically consistent with Klein's group theoretical understandings.

The connections of the essential concept of geometry with groups also extends to Simplicial Homology and to Noncommutative Geometry.

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