I just made some intellectual insults, thought it might be helpful

Hahaha

It doesn't really make sense?
Something like "Your closure is the entire space" or "there is only one closed set that fits around you" would be better imho.

@TempestasLudi although by accident I was [procrastinating while] studying topology at the time, I had the model theoretic notion in mind. Either way you should be dense to be able to satisfy the density axiom 😆

@beleg But... How does density have anything to do with embedding the real number line?

@TempestasLudi Embedding here is something that preserves structure, so the real line (the structure of reals and their usual ordering) being embedded into some other structure M mean that it is (order-) isomorphic to some submodel of M, and therefore they are elementary equivalent. That means the truth of any formula, as well as the density axiom (for all x and y where x<y there is some z such that x<z<y) carries over from the real line to M.
I don't know (yet!) but I guess you can translate all this to the topological definitions of embedding and density.

Oooh. You're talking about the density of betweenness.
The thing I know as "dense", works as follows: A subset S of a topological space (X, t) is dense if its closure equals X, i.e., if X is the smallest closed set (in (X, t)) that contains S.

@TempestasLudi I'm just learning topology, so I'm not sure but I guess these notions coincide in a metric space.

Hm... On the open interval (0, 1) (subset of R), betweenness is dense, but (0, 1) is not dense in R.