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It seems like alot of the counter examples in theorems about metric spaces occur in spaces with the discrete metric.

Examples:

1) A totally bounded subspace $A \subset M$ is bounded but the reverse is not true if you look at the discrete metric on integers.

2) A sequentially compact subspace is closed and bounded but the reverse isn't true if you consider the same space from example 1) (you can find a sequence of integers that doesn't have a convergent sub sequence).

I know that the more general the theory the better, but including the discrete metric brings alot of restrictions. What makes the discrete metric so important?

John Cataldo
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    It's (essentially) the metric of any finite set, and of the integers, at least as far as convergence and continuity is concerned. I'd say those are pretty important, wouldn't you? – Arthur Jan 30 '18 at 11:59

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The discrete metric is an extreme example which makes it easier to highlight phenomena that can in fact occur in much more common cases. The fact that we use this particular example does not mean that you have to be so extreme to find counterexamples to your favourite internal pictures of how things work.

Alternatively, you can design your own axioms that will exclude such extreme cases and allow you to derive nice properties. That is also a common in mathematics.

Arnaud Mortier
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