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Someone asked for an example of an infinitely uncountable set with only isolated points.

Someone answered $\mathbb{R}$ with the discrete topology and many agreed.

Then I read that: Any discrete subset of Euclidean space is countable, since the isolation of each of its points (together with the fact that the rationals are dense in the reals) means that it may be mapped $1-1$ to a set of points with rational coordinates, of which there are only countably many.

Which seems to contradict the example?

This is a very basic question but i seem to misunderstand something?

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The two examples have different topologies. – André Nicolas Mar 20 at 4:10
The discrete topology on $R$ is not a subspace of the usual topology on any $R^n$. Check the def'n of subspace, as opposed to subset. – user254665 Mar 20 at 4:26
You need to be more discrete. – copper.hat Mar 20 at 4:27
"Uncountable" implies "infinite," so "infinitely uncountable" is superfluous. – Akiva Weinberger Mar 20 at 4:50
.... the rationals are not dense in reals with the discrete topology. But basically, R with discrete topology is not at all a subspace of euclidean space at all. – fleablood Mar 20 at 5:45
up vote 9 down vote accepted

The sentence

Any discrete subset of Euclidean space is countable

assumes that we are using the usual topology - that is, "Euclidean space" means "$\mathbb{R}^n$ with the standard topology." The standard topology is, of course, not discrete.

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Yes, of course! LOL...its to late for doing math... – JKnecht Mar 20 at 4:13

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