So, I just did a counterexample in topology that relied upon a well ordering in which you there were a pair of elements that you could not connect between in a finite number of steps. (The standard odds then evens, or evens then odds ordering of the natural numbers). I believe the statement would have been true if my well ordering had an additional axiom that this was impossible. Does such a thing have a name/been studied before/have any significance?
To be precise, I'm looking for a term for a well ordering operation on a set that has the additional property that between any two elements $x<y$, with a finite number of applications of the successor operation on the lesser element, you can reach y from x.