The following examples are our Target i.e. when elaborating/working with the Axioms and the question arises whether to accept or reject a new formulation, I check whether all examples are still covered or not (sometimes by slight modifications of the example without changing its essence).
Example 1 Real Line as Q-Space
Example 2 Circle-Group as Q-Space
Example 3 Genesis as Q-Space
The solid circles are the time-like geodesics, the dashed triangles are the non-time-like geodesics. Not indicated are the additional non-geodesic time-like lines like {Night, Morning, Before Noon, Noon, Afternoon, Evening} and their symmetric partners. Boxes are open, Circles are closed in O-Topology. Shadowed one Normal set.
Example 4 Simple Discrete Wave as Q-Space
The solid lines are the time-like geodesics, the dashed triangles respectively lines in different colors are the non-time-like geodesics. Shadowed one Normal set. The Simple Discrete Wave represents in a way the unfolding in time of Genesis.
Example 5 Simple Discrete Wave with Spin as Q-Space
The solid lines are the time-like geodesics, the dashed lines are the non-time-like geodesics. Shadowed one Normal set.
Example 6 Complex Plane as Q-Space
Example 7 Quaternion-Space as Q-Space
Example 8 Complex Plane as refined Q-Space
Example 9 Quaternion-Space as refined Q-Space 
Example 10 Orientable Lorentz-Manifold as Q-Space
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