We had again to modify the definition in two aspects: (1) Taking into account,as we did before when defining the topology that not all elements are places, and (2) introduce some means to connect the locally defined linear structures.

While the Light-cones are the underlying combinatorial structure from General Relativity to model the Causal (or Conformal) Structure, it seems that *totally normal neighborhoods* serving as *charts* and the *atlas* formed by a subset of the class of *totally normal neighborhoods* 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. In the second intent, I limited the definition to a pure local

**yet this way I lost the connection from local context to local context, which for manifolds is naturally provided by their atlas of charts.**

*G*#### 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) Defines as strictly *local *relation ** G **within a base-set.

(4) Define 3 properties required to make

**a useful Local Linear Structure:**

*G*Before continuing, remember that in our topology not necessarily all elements are closed. We singled out one set, the set of

*as not-open, while all others are open.*

**P**laces(4.1) may be called the

*line*property: 2

*places*define a

*line*i.e. if two different elements

**c**and

**d**are collinear to the same

*places*

**a**and

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

*line.*

(4.2) may be called the

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

*line*and any

*line*contains at least two

*places*.

(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

*places,*two

*places*define exactly one

*line*and finally any 2 different elements

*are on at least 2 different*

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

*lines*with each at least 3 elements.

(5) defines

*linear*sets i.e. those that with any 2

*places*contain all

*locally*collinear elements. The <

*linear set*

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

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

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

*linear independence.*

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

*lines.*

The predicate (7) connects

*lines*to the previously topologically defined

*paths*: all

*lines*are

*paths*. This turns

*lines*and hence the

*local linear structure*into a topological invariant. The predicate more over requires that the whole linear structure may be

*densely generated*at each element that is that there is a

*countable set of lines*all containing this element, which

*generates*the whole neighborhood.

(8) defines

*Normal Blocks*as equivalent for totally normal neighborhoods i.e. a

*block*

**and**a

*local linear structure*, right as the definition of a totally normal neighborhood requires a specific affine connection .

(9) introduces the base set for

*global*, which –as it appears in (10)- is nothing else but the another expression of forming an

**G***atlas*out of totally normal neighborhoods.

(10) defines the class of

*global*, formed taking subsets of

**G***Normal Blocks*. Such a subset has to meet 3 conditions: (10.1) its

*blocks*have to cover the whole set, where the

*global structure*is just the join of the

*local linear structures*. (10.2) The local structures have to be

*compatible*, i.e. agree on shared subsets and finally (10.3) the whole subset of

*normal blocks*has to be connected by means of

*local linear structures*with shared subsets. Actually (10) was modeled following the idea of an

*atlas*

*of compatible charts*for a

*manifold*.

(11) Define the class of Q-orders that allow such a

*global*and the class of these.

**G**(12) Simply asks that this class is not empty.

Going back to *Totally* *Normal Neighborhoods *of Lorentz-Manifolds. All points in a *Totally* *Normal* *Neighborhood *are connected by at most one *geodesic, *whence different points differ at least in one *geodesic*. A *geodesic* relies in its definition on the affine connection. Obviously *lines* are *paths* in the underlying topology. In case of standard Lorentz-Manifolds, just 4 *lines* are sufficient to generate the whole neighborhood. *Totally Normal Neighborhoods *define *charts, *which in turn for a fixed affine connection define jointly an *atlas*, which obviously is connected for connected manifolds.

*Voilá *exactly what the axiom-set 3 claims for *Linear Q-orders*.