Surrogate Reals
Topology often refers to the reals which is the normal path to topology.
The reals are generally taken to be the completion of the rationals, which are pairs of integers.
There is another easy way to get to the topological reals: the completion of any simply ordered denumerable set where between any two members there is a third and there is no first nor last member.
This is sort of an axiom set for reals.
This is all Berkeley circa 1954.
Perhaps this short note dominates this page.
We develop here a characterization of the reals for topological purposes.
Here lower case letters denote points in a space.
Upper case letters denote sets of those points.
In the following lower case letters denote reals and upper case denotes sets of reals.
There is a relation ≤ between the reals such that:
- non-empty
- ⋁x(x=x)
- ≤ is transitive
- (∧x∧y∧z (x≤y & y≤z → x≤z))
- ≤ is total
- ∧x∧y (x≤y or y≤x)
- ≤ is reflexive
- ∧x (x≤x)
- ≤ is antisymmetric
- ∧x∧y (x≤y & y≤x → x=y)
- ≤ is dense
- (∧x∧y ((x≤y & x≠y) → ⋁z(x≤z & z≤y & z≠x & z≠y)))
- ≤ is endless
- ∧x(∨y(x≤y) & ∨y(y≤x))
- ≤ is complete
-
∧E((∧x∧y(y∊E & x≤y → x∊E))&∨x(x∊E)&∨x(¬x∊E) → ∨x(∧y(y=x or (y≤x ↔ y∊E))))
If a total ordering is dense, complete and endless then it is order-isomorphic with the reals.
That means that there is a one to one map (bijection) between the points and the reals which preserves ordering.
This isomorphism proves the compatibility of these requirements, and also that the requirements are categorical.
An open interval for an ordering is, form some two distinct points x and y the set of points z such that x≤z & z≤y & z≠x & z≠y.
The set of open intervals form a basis for the normal topology of a total dense endless complete.