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"Densest" is not really an applicable term here. I take it you mean "has the highest cardinality?" In this cast there are an infinite number of these. A theorem states (I forget the name) that a subset of a set can have at most the same cardinality of that set. So we need a set S such such that S ⊆ ℝ and |S| ≈ |ℝ|. Like I said, many sets fit this description, i.e. ℝ itself, any open or closed interval on ℝ like [1,16) or (-∞, 3), any union of any subset of ℝ and an open or closed interval on ℝ such as (12, ∞) ∪ {e}. I suppose that there are many types that I may be forgetting, but I hope you understand. =]

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Q: What is the densest subset of real numbers?
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