Partial obsolete – see below

This post will be dedicated to 1-dimensional Q-Orders, studying the effects Axiom **6 Q-Topology** has on them and preparing the ground for the n+1-dimensional case. **Axiom 6 **

We will study 4 cases: the Rational Numbers, the Real Numbers, the Cyclic Group Zn and the Circle Group, using for the latter geometric representations on the Unit-Circle when appropriate.

Here are the more formal definitions: **Lemma F 6 **

The proof for F 6.2-3 –we assume the normal order <, was already given –they are as full orders also partial orders-, for F 6.7-8 the reader, if in doubt, is kindly invited to go back at least to Going in Circles – Part III. The constraints -Reduced, Regular, Q-Connectivity- are obviously satisfied.

Obsolete by the Great Simplification … yet still useful for heuristics .. and will be replaced soon.

Next we will have a look on the corresponding *L*ine-Sets. **Lemma F 7 **To show the intended above:

Please note that in all cases the red or blue marked sets designate or represent just

**one**line, despite that apparently this line is not always connected.

Next we will look at l-closed points, and the open line-filter and open neighborhood-filter for each case. **Lemma F 8 ** As the definition for open L-intervals –hence the open neighborhood-filter- requires that only for l-closed points there is a surrounding L-interval, in case of

**Z**n any interval is by definition open, as all points are open hence any neighbor-hood is open. Different in the other three cases.

Next a look on the topologies and whether they fit into Axiom 6. Nothing new, all rather very basics from elementary topology. **Lemma F 9 **No big surprises yet, as it is no surprise that Real Line and Circle Group both satisfy Axiom 6.7, mainly due to their Dedekind-Completeness.

**This end our first round through 1-dimensional Q-Orders.**

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