Consider the set $E=\mathbb{Q}\cap[0,1]$, and let $\{q_{j}\}_{j=1}^{\infty}$ be some enumeration of this countable set. For every $\epsilon>0$, the cubes $\{Q_{j}\}_{j=1}^{\infty}$ of length $\ell_{j}=\frac{\epsilon}{2^{j}}$ centered at each $q_{i}$ clearly cover $E$, and we have $\sum_{j=1}^{\infty}|Q_{j}|=\epsilon$. This implies that $m(E)=0$.
This example is typical. But here's my question. The closure of $E$ is $\bar{E}=[0,1]$. In particular, $\bar{E}\backslash E=\mathbb{I}\cap[0,1]$. If we examine our cover, we see that to each $q_{j}$ we have placed a neighborhood of radius $\frac{\epsilon}{2^{j+1}}>0$. Because $A$ is dense in $[0,1]$, for every $x\in[0,1]$, $B(x;\delta)\cap A\neq\emptyset$ for any $\delta>0$.
In other words, the collection of balls $\{B(q_{i};\frac{\epsilon}{2^{j}})\}_{j=1}^{\infty}$ ought to cover $[0,1]$. But if it did, then it would contradict what was shown above.
How do you resolve this (for general countably dense subsets).
This question arose from a related problem on a post I made regarding a question as to the relationship of outer Jordan measure a set $E$ and its closure $\bar{E}$, and can be found here: Closure, Interior, and Boundary of Jordan Measurable Sets.. In particular, I want to know if it's possible to justify "$\epsilon$-fattening" a cover of $E$ to a cover of $\bar{E}$. In other words, can we take a finite cover $\{Q_{j}\}_{j=1}^{N}$ of $E$ by cubes of length $\ell_{j}$, and fatten each cube by no more than $|Q_{j}\leq|Q'_{j}|\leq|Q_{j}|+\frac{\epsilon}{N}$ and then claim $\{Q'_{j}\}_{j=1}^{N}$ is a cover of $\bar{E}$? Apparently this should be true for finite covers, but it's clearly not true for countable covers because of the obvious counter example presented at the beginning. But my justification for why it's true for finite covers carries over to countable covers (in that $\bar{E}\backslash E$ is a collection of limit points of $E$, so arbitrarily fattening up a cover should capture these points, hence cover $\bar{E}$).
There must be something wrong with my thinking here, and I really need to get it resolved! So much thanks in advance to anyone who can help me with this!