Characterize RTop and its subspaces

[Columbus240]

There are multiple ways to characterize RTop and its subspaces (up to homeomorphism) based on order or metric properties. Such theorems could help transport topological properties.

IIRC the following statements are true:

• Every order-complete, linear-order containing a countable and dense subset, is order-isomorphic to Rle on a closed interval.
• Every non-empty, unbounded, linear order with the least-upper-bound property and a countable and dense subset is order-isomorphic to R.
• Every unbounded, countable, dense linear order is isomorphic to Q.
• Isomorphic orders induce homeomorphic topologies.
• All open/closed intervals/rays in RTop are homeomorphic. By showing that each map x => a*x+b for a<>0 is an automorphism of RTop and
• Homeomorphisms induce homeomorphisms on subspaces.

I'm sure there's some way to use the metric structure, but haven't yet looked into a precise statement. Metrically complete and separable are probably important again.

Small conjecture: an order topology is separable/second-countable iff there's a countable dense subset. A subset A of an ordered set X shall be called dense if for all x, y : X with x <= y there exists a : A such that x <= a <= y.