When viewed as a metric space under the metric d(x, y) = |x - y|, the naturals and the rationals are fairly different. No matter how small d(p, q) is, there’s a rational w such that d(p, w) < d(p, q). Clearly, the naturals don’t have this property. Is there a name for this property? I’m thinking self-dense, but I have no idea whether that’s standard terminology.
I call this property “Fred”. 
I think this is another aspect of non-countability.
Both of those sets are countable. And on the reals, you could define d(x, y) = ceiling(|x - y|), and you’d have the same contrast.
Sounds like it’d be similar enough to ‘densely ordered’ as to not need distinction. Not that I know enough terminology to know if it truly has a specific term.
Yeah, but I never said that I’m imposing an ordering on these sets. Just because the naturals and rationals usually have an ordering associated with them doesn’t mean they always have to.
Suppose I replaced the reals in my last example with continuous functions on [0, 1]. Then I use the sup-norm and the ceiling of the sup-norm as metrics. One has the property, and the other doesn’t, but there’s no (total) ordering.
Oops. I realized that as soon as I posted … we’re talking about metric spaces here. They don’t necessarily have to be ordered, do they? (Weird if they’re not, but that’s half the fun.)
Or exactly what you said on preview
‘Self-dense’ seems a little too generic to me, though it might work. ‘Metricly dense’ is too awkward. A ‘dense metric’ sounds backward from what you’re saying. How about ‘close’? (evoking some element of the metric there, but it unfortunately sounds too much like ‘closed’). “Closely spaced?” way too confusing.
If you generalise metric spaces to topological spaces:
A subset H of a topological space T is (everywhere) dense in T if Cl(H) = T.
Cl(H) is the closure of H defined as the union of H and all the limit points of H.
(Given a point x and a subset H of a toplogical space T, x is a limit point of H if every open set containing x contains some point of H other than x.)
Now the rationals form a dense subset of the reals because any point x in R (the set of the reals) is a limit point of Q (the set of the rationals), which I’m pretty sure is a corollory of your original property. I think if you replace topological space with metric space and a similar defintion for limit point under metric spaces the same logic applies
Hope this is clear.
I don’t think that’s quite what I’m looking for. Go back to the two metrics I gave on C[0, 1], and consider that without looking at that set as a subset of a larger metric space. In one case, the space is dense in itself, and in the other it isn’t.
hmm ok I’ll have a think…
It may be that “dense in itself” is the best name out there.
What this came out of, for those who are curious, is a riddle from William Wu’s riddle page:
[quote]
Find the error in the following argument:[ol][li]d/dx(x[sup]2[/sup]) = d/dx(x + x + … + x) (x terms)[]d/dx(x + x + … + x) (x terms) = d/dx(x) + d/dx(x) + … + d/dx(x) (x terms)[]d/dx(x) + d/dx(x) + … + d/dx(x) (x terms) = 1 + 1 + … + 1 (x terms)1 + 1 + … + 1 (x terms) = x[/ol]Therefore, d/dx(x[sup]2[/sup]) = x.[/li][/quote]
Of course, the error is that the first step is only valid if you restrict yourself to natural numbers, and since that metric space isn’t dense in itself, there’s no such thing as a limit there anyway.
I was just curious if there was a shorthand name for that property.
Its an interesting problem, I’ll have a look at it, but as far has I can remember you can only use d/dx if you are working with intervals in the reals, so the property you give isn’t sufficient for differentiation. Anyway, I’ll have a look.
The property is called (or described, if you prefer), “no isolated points”. More precisely, the integers are discrete (metrically discrete which is somewhat different from topologically discrete) while the rationals have no isolated points.
If you are curious, the set of 1/n, for integers n, is topologically discrete, but not metrically (or uniformly) discrete.
Hari: could you explain in some greater detail the difference between topologically discrete and metrically discrete?
Well, that’s funny; all this time I thought Naturals were rather indiscreet…