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Given the standard topology, is there any relationship between dense, uncountable sets and isolated points? For example, the set of irrationals is both dense and uncountable and contains no isolated points.

Just something I've been wondering working my way through real analysis.

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is there a restricted context to this question? eg, are we just talking about $\mathbb{R}^n$? – Dustan Levenstein Mar 22 '12 at 20:08
Just $\mathbb{R}$, $n=1$ – docjay Mar 22 '12 at 20:10
up vote 3 down vote accepted

First of all, if a set is dense, it has no isolated points.

A good example for a set that is uncountable, with no isolated point and not dense is the standard Cantor set.

Any open set is uncountable, but has no reason to be dense. You still can add a point so it has an isolated point.

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