There is something strange (I think) about the complete regularity separation axiom. Consider the definitions.
- T0 means for every two distinct points there is an open set containing exactly one of them.
- T1 means for every two distinct points there is an open set containing one (specified) but not the other.
- Hausdorff means for every two distinct points there are disjoint open neighborhoods around them.
- Regular means for every point and closed set not containing it there are disjoint open sets containing them.
- Normal means for every two disjoint closed sets there are disjoint open sets containing them.
However when we get to complete normality we have the following definition:
- Completely regular means for every point and closed set not containing it there is a continuous real-valued function which is 0 at the point, and 1 at every point in the closed set.
It seems odd that all the other separation axioms are easily defined in terms of the open (and maybe closed) subsets of that space but for complete regularity you need some "external object" (the real line with the metric topology). Is there a way to define complete regularity without using an "external object" like this?