Saturday, January 16, 2010

Q-Topology – I

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

Theorem 1 Q-Topology

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

T 1.1-3 Intervals and Sub-Cone Neighborhoods
Eqn17Intervals J are complete pieces of lines, i.e. contain all elements of a line that are between some border-elements. [Closed] Points 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
Eqn18 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.

Lemma T 1.5 Meet and Join

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