After slashing the original definition of Axiom 6, here comes a similar reduction of Axiom 7, originally introduced in The trapped Arrow of Time – Part III.

The new **Axiom 7 Q-Loops **

**Axiom 7.1**are technical definitions:

*completely Q-ordered*sets and the

*closed hull*of a set.

**Axiom 7.2-4 **introduce a substitute for Jordan-Curves. 7.2 defines the property of *being connect *for a set in terms of topology. 7.3 defines the relation of being topologically separated for two points. Please note that we don’t require that the whole space is T1. 7.4 defines * J-*Curves as sets that contain for each point at least one

*separated partner*and

*fall apart*

**exactly**if a

*separated pair*is removed. This definition requires implicitly the Axiom of Choice, therefore it’s flagged.

For the moment let’s assume that * J-*Curves exist -later on we will claim the existence of rather specific ones- and see whether the definition meets our expectations.

**Axiom 7.5**introduces the set of

*connected subsets*into which a separated pair splits the

*J*

*-*Curve. By definition for

*Curves, there must be at least 2 of them.*

**J**-**Lemma S 2 Segments**

The lemma shows that –as intended- a separated pair splits a

*J*

*-*Curve in just 2 segments. And these 2 segments contain at least each a point for a separated pair, that is we’ve got two pairs that mutually separate each other.

Sounds familiar? Well, be aware that the term * J-*Curve was introduced and some of it’s properties shown

*without*reference to the initial Q-Order, yet –as intended- on a

*Curve there exists a natural Q-Order, as defined by*

**J**-**Axiom 7.6**.

**Lemma S 3 J-Order**

Be aware that by no means all * J-*Curves correspond to q-ordered sets, as shown below.

The above yellow Rhombus is a

*Curve, yet it’s not a Q-ordered set but rather built by of 2 Q-ordered sets, the left and right side.*

**J**-It appears as if we might start out just with some topology with some nice properties … and [re-]construct the Q-Order. For the moment we ask only –consistent with our whole approach- that every *totally q-ordered* set shall be consistently embeddable into some * J-*Curve, where consistency means that original Q-Order and derived Q-order of the curve are the same.

**Axiom A 7.7 Consistent Embedding **

Axiom A 7.7 has backward consequences for the Q-topology. **Lemma S 4 Connected Space **As immediate consequence of Axiom 7.6,

*Curves finally*

**J**-*do exist.*

*Curves connect the whole space (S 4.2), which is hence a connected topological space (S 4.3).*

**J**-Finally **Axiom 7.8 Local Orientation **

This Axiom establishes an intrinsic relation between * J-*Curves –remember they are closed- and the underlying Q-Topology. For each point –respectively its

*closed hull*- there shall exist at least one neighborhood, sufficiently large that it can be split into two subsets –sometimes called local future and local past, or local input and local output- such that each of these subsets is

*J-convex –*i.e. any two points can be connected by a

*Curves, but sufficiently small that any*

**J**-*Curve that connects between the sets contains at least one external point.*

**J**-The picture below illustrates the concept.

The two sets are

{ {(0,0),(-1,0)}, (-1,0), {(-1,0),(-1,1)}, (-1,1), {(-1,1),(0,0)}, {(-1,1),(0,1)}, (0,1),{(0,1),(0,0)} }

{ {(0,0),(0,-1)}, (0,-1), {(0,-1),(1,-1)}, (1,-1), {(1,-1),(0,0)} {(1,-1),(1,0)}, (1,0),{(1,0),(0,0)} }

**Axiom 7.8** requires at least two dimensions (or two * J-*Curves). As to be shown, it captures the underlying essence of the Hawking-construction for

*regular curves,*which in the original text is scattered between local properties of the manifold –existence of local convex neighborhoods in terms of the Manifold-Topology, global causality-conditions –strong causality-, all needed to effectively define Regular Curves, and finally the properties defined by the construction as such.

**This ends our preliminary presentation of the new Axiom 7 Q-Loops.**

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