# Permitted value of epsilon in Discrete Metric Space

If we define a Metric Space with Discrete Metric, say $(\mathbb{R},d)$. Then whenever we talk about epsilon or delta, such as talking about neighbouringhood, limitpoint, can we take epsilon to be not discrete, or does it have to discrete by discrete, I mean {1,0}?

If I take a subset {1,2,3,4,5}, is it that the set is compact but has no limit point? Is it different?

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I realize this is pedantic, but discrete and discreet mean different things. I have edited your post to reflect this. Sorry again for the nitpicking. – Potato Jan 9 '13 at 8:42
What do you mean by taking discrete/non-discrete epsilon? – Ilya Jan 9 '13 at 8:49
Sorry, I meant only 1 and 0 – 007resu Jan 9 '13 at 8:52

## 1 Answer

If you define a discrete metric space as a space where the metric has range $\{0,1\}$, you can do with epsilons from the set $\{1/2,1,3/2\}$. But you are not prohibited from using other nonnegative numbers too.

Every nonempty set in a metric space has a limit point, namely all points in the set. But it cannot have an accumulation point if it contains only finitely many points. You can also show that no set has an accumulation point in a discrete metric space, since every point has a neighborhood containing only that point itself.

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Is there a difference between accumulation point and limit point in metric space? – 007resu Jan 9 '13 at 8:57
@Freddy: what is the definition of the accumulation point in general? – Ilya Jan 9 '13 at 9:01
Usually, $x$ is a limit point of $S$ if every neighborhood of $x$ contains a point from $S$ and an accumulation point of $S$ if every neighborhood of $x$ contains a point from $S$ different from $x$. This seems to be standard nomenclature, but I have seen a few exceptions. – Michael Greinecker Jan 9 '13 at 9:04
Well I was not aware of them being different and I felt limit point is the one with every neighbourhood of x containing a point from S different from x. – 007resu Jan 9 '13 at 9:09
@Freddy It is possible that you learned from a text using that convention. – Michael Greinecker Jan 9 '13 at 9:11