1

In a popular maths book I find this sentence, in the context of an explanation of the difference between discrete and continuous, especially as regards groups:

The group of integers is discrete; that is to say, its elements do not combine into a continuous geometric shape in any natural sense.

I have no problem with the sentence in itself, within the limits of a popular exposition. I was just asking myself: Is there a “non-natural”, but non-trivial, sense in which integers can be seen as forming a continuous “geometric shape”?

DaG
  • 486

2 Answers2

1

There are countably many integers ($\mathbb{Z}$) and rationals ($\mathbb{Q}$), and hence a 1-1 correspondence between the integers and the rationals. The rational numbers are dense in $\mathbb{R}$ so you might be able to think of $\mathbb{Q}$ as a "line", and by countability also $\mathbb{Z}$.

pshmath0
  • 10,565
  • 1
    Perhaps a very "hole-y" line. You could also map it to the unit circle by a bijective argument with $\mathbb{Q}$ again. Let $\frac{p}{q}$ be a rational number in lowest terms. Consider the map $\frac{p}{q}\mapsto e^{2\pi ip/q}$. If I recall correctly, this is a dense subset of the unit circle. Again, this is has holes but you can't get around that. – Cameron Williams Feb 11 '14 at 18:47
0

The concept of continuity I think you're trying to express is captured by the topological concept of connectedness. There do exist countable connected spaces (even Hausdorff ones), but most constructions are odd. If you applied something like this to $\mathbb{Z}$ there wouldn't be a good notion of distance like there is in the standard topology of $\mathbb{Z}$ (every countable metric space is disconnected).