The structure (Q;Q) as defined by Axioms I-VII enables the definition of a Topology for Q.

We will present this theorem, its definitions and proofs step by step.

**T 1.1-3 Intervals and Sub-Cone Neighborhoods**

Intervals J are complete pieces of lines, i.e. contain all elements of a line that are between some border-elements. [Closed] **P**oints are elements that appear as limits. (see Lines, Cuts and Dedekind for details). An element is considered *truly inside* a set if this set contains at least two neighboring elements. We continue defining for an element the set of Intervals it’s truly inside. And finally a type of *Sub-Cone Neighborhoods* V, sets which contain for each element of the Interval-set of an element at least one representative; *Sub-Cone Neighborhoods* as –seen from GRT- they contain the initial part of the corresponding light-cone (minus its light-borders).

**T 1.4-5 Topology ** As sets for the Topology O we define all those sets, which for each

**closed**point p contain at least one of the defined Sub-Cone Neighborhoods. The Lemma T 1.5 simply states that O is a topology; i.e. that intersections and arbitrary joins of

**open**sets are open.

** **In a next post we will have a look on some properties of this Topology.