While the Light-cones are the underlying combinatorial structure from General Relativity to model the Causal (or Conformal) Structure, it seems that Normal Neighborhoods and Normal Coordinate-Systems (alas the Inertial-Frames or the heuristic base for the Einstein Equivalence Principle) are the proper candidates upon which to model the combinatorial equivalent for the Projective Structure.

In my last version for the Q-Space Axioms, I committed a terrible conceptual mistake, which for its detection and clean-up took some time. Mislead by the appearance of the concept of geodesics as if something *global *yet based on only *locally *defined properties, I introduced a similar global relation ** G** with local properties.

Now

**will be a strictly**

*G**local*relation with only

*local*properties.

Here is the new Axiom Set 3 titled

*Linear Q-Spaces.*

#### Axiom Set 3 Linear Q-Spaces

(1) As before, take Topological Q-Spaces as departing point. (2) As before introduces the topological closure and the closed hull of Alexandrov-Sets (=Double Cones) as *base-set*.

(3) However defines ** G **as a strictly

*local*relations within a base-set.

(4) Define 3 properties required to make

**a useful Local Linear Structure:**

*G*(4.1) may be called the

*line*property: 2

*points*define a

*line*i.e. if two different points

**c**and

**d**are collinear to the same points

**a**and

**b**, then they are on the same

*line.*

(4.2) may be called the

*separation*property: different points differ in at least one

*line.*

(4.3) may be called the

*connectivity*property: it’s possible to get from anywhere to anywhere in a final number of

*linear*steps.

The combination of (4.1) and (4.2) imply that locally

**satisfies the axioms for Incidence geometry: all**

*G**lines*have at least 2 points, two points define exactly one

*line*and finally any 2 different points are on at least 2 different

*lines*, in our case even stronger: 2 different

*lines*with each at least 3 points.

(5) defines

*linear*sets i.e. those that with any 2 points contain all

*locally*collinear points. The <

*linear set*

*generated by a set of points*> is the smallest

*linear set*that contains them all. The concept of a

*generated linear set*allows to introduce a combinatorial concept of

*linear independence.*

(6) introduces the set of 1-dimensional linear sets or

*lines.*

(7) defines a set of

*Frames*i.e. minimal subsets of

*mutually linear independent lin*es (7.2), which generate the whole set (7.1) and are fixed at some point (7.3).

(8) Is a later required

*technicality*: a

*Frame*should be countable at least.

(9) singles out the

*anchor-points*for

*Frames.*

(10) defines for a Q-Order a

*Locally Linear Structure*(though without metric part yet): all those pairs of a base-set

**B**and a relation

*defined on B such that it defines a*

**G***linear structure*on

**B**(10.1), every

*line*is at the same time a

*path*in the underlying topology (10.2), all

*frames*are at least countable (10.3) and finally there is at least one anchor point (10.4). Please note that there might be more but only one

*i.e. the same base-set may carry more but one*

**G***linear structure*. Finally a

*Locally Linear Structure*is a

*topological*invariant, as a

*local homeomorphism*takes

*lines*to

*lines*(consequence of 10.2).

(11) Selects all those

*Q-Orders*for which there is a corresponding

*LLS*such that there is a

*countable subset*, whose anchor-points cover the whole set. The final

*LLS*is just the union of LLS for all

*Q-orders*.

(12) Just claims that there are Q-Orders with a

*Locally Linear Structure*

Going back to *Normal Neighborhoods *of Lorentz-Manifolds. These are defined by selecting first one anchor-point (in the Einstein-concept the famous free-falling-observer). All points in a *Normal* *Neighborhood *of this anchor-point are connected by at most one *geodesic, *whence different points differ at least in one *geodesic*. All points connect by means of a *geodesic *to the anchor-point, and finally there is at least one *frame –*the *normal coordinate system* at the anchor-point- that generates the whole set. Obviously *lines* are *paths* in the underlying topology. Finally there is a *Normal* *Neighborhood *for any point. The *dimension* (8) for *Frames* is just 4. And the second-countability for Lorentz-Manifolds implies the countability restriction in (11) *Voilá *exactly what the axiom-set 3 claims for *Linear Q-orders*.

Next step will be to explore the *combinatorial elements *of *connections* when expressed as *mappings of frame-sets*.