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I'm curious about the notion of a minimally dense subset of a given space (I've been using $[0,1]$, but if others are interesting, I'm interested in that too!). Here are two questions.

  • Does there exist a set $A$ that is dense in $[0,1]$, but for all infinite $B\subset A$, $A\setminus B$ is not dense in $[0,1]$? I suspect that the answer to this question is no, but I haven't been able to prove it. I'm also curious about what modifications to this statement can create a notion of "minimally dense" subsets.

  • For all $\alpha\in \mathbb{R}\setminus\mathbb{Q}$, there exist infinitely many rational numbers $\frac{m}{n}$ with $|\alpha-\frac{m}{n}|<\frac{1}{n^2}$. In his analysis book, Browder remarks that this theorem is a way that one can quantify density. Is there a way to use statements like this to come up with a "minimally dense" subset of [0,1]?

Let me know if there are other ways of talking about this notion, or whether you think there is something wrong with the idea of a minimally dense set in general. Thanks!

unit-interval
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No, if $A$ is dense in $[0,1]$ then there exists an infinite set $B\subset A$ such that $A\setminus B$ is dense in $[0,1]$.

For example, say $(a_n)\subset A$ is a sequence with $a_{n+1}<a_n$ and $a_n\to 0$. Let $B=\{a_{2k+1}\}$. Then $A\setminus B$ is dense:

Since $a_{2k}\to0$, $A\setminus B$ contains a sequence that converges to $0$. Suppose that $x\in[0,1]$ and $x\ne0$. Choose a sequence $(b_n)\subset A$ with $b_n\to x$. Since $a_n\to0$ and $b_n\to x\ne0$ it follows that $B\setminus A$ contains all but finitely many of the $b_n$; hence $A\setminus B$ contains a sequence converging to $x$.

David C. Ullrich
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