# 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

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.
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