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If we partite $\mathbb{R}\backslash\mathbb{Q}$ as $\cup_{i\in\mathbb{N}}A_i=\mathbb{R}\backslash\mathbb{Q}$, $A_i\cap A_j=\emptyset$ if $i\ne j$, can it hold that $A_i$ has no rational limit point?

Actually, we have irrational perfect sets, but they don't seem to be able to contain much of $\mathbb{R}\backslash\mathbb{Q}$. Or if we can express the irrationals as a disjoint union of such sets, then we are done. But this too seems hopeless.


Furthurmore, does it hold that $\cup_{i\in\mathbb{N}}\bar{A_i}=\mathbb{R}$?

Actually, this is a strengthening of the first problem. The first one seeks to prove that whatever partition we make, the outcome cannot be too coarse. The second one further discusses whether it is still dense.

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No, it cannot hold.

Your problem asks for a family $\{A_n\}_{n\in\mathbb N}$ such that $A_i'\cap\mathbb Q=\emptyset$ for all $i\in\mathbb N$, but this means $\overline{A_i}\cap\mathbb Q=\emptyset$, as $\overline{A_i}=A_i\cup A_i'$ and $A_i\subseteq\mathbb R\setminus\mathbb Q$, but this is equivalent to $\mathbb Q\subseteq \overline{A_i}^c=int(A_i^c)$ for all $i$. Thus your problem, ignoring the disjointedness part, is equivalent to finding a family $\{A_i\}_{i\in\mathbb N}$ of sets of reals such that $\bigcap_{i\in\mathbb N} A_i=\mathbb Q$ and $\mathbb Q\subseteq int(A_i)$ for all $i$, but this cannot happen because of Baire's First Category Theorem:

Let $\mathbb Q=\{q_i:i\in\mathbb N\}$ be an enumeration of $\mathbb Q$. As $\mathbb Q\subseteq int(A_i)$ for each $i$, each $int(A_i)$ is open dense in $\mathbb R$, then clearly each $int(A_i)\setminus\{q_i\}$ is open dense in $\mathbb R$; as $int(A_i)$ either contains an open interval that contains $q_i$; in such case $q_i\in \overline {int(A_i)\setminus\{q_i\}}$, or $q_i\notin int(A_i)$. Therefore $\bigcap_{i\in\mathbb N}(int(A_i)\setminus\{q_i\})$ is dense in $\mathbb R$, by 1. However $\bigcap_{i\in\mathbb N} int(A_i)\subseteq\bigcap_{i\in\mathbb N} A_i=\mathbb Q$, which implies $\bigcap_{i\in\mathbb N}(int(A_i)\setminus\{q_i\})=\emptyset$. Contradiction.

Now, if you ask for $\bigcup_{i\in\mathbb N} \overline{A_i}=\mathbb R$, it's even easier, as this means $\bigcap_{i\in\mathbb N} int(A_i^c)=\emptyset,$ however, $\mathbb Q\subseteq int(A_i^c)$ for each $i$, in consequence each $int(A_i^c)$ is open dense, so you obtain a contradiction using 1.

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  • $\begingroup$ It's actually $\overline{int(A)}\cap\mathbb{Q}=\emptyset$. It implies $\overline{A}\cap\mathbb{Q}=\emptyset$ in this case though. $\endgroup$ – Anonymous Coward Nov 11 '13 at 6:25
  • $\begingroup$ @AnonymousCoward what do you mean? $\endgroup$ – Camilo Arosemena-Serrato Nov 11 '13 at 13:07
  • $\begingroup$ I mean we should exclude the isolated point (i.e. consider the derived set, not the closure). $\endgroup$ – Anonymous Coward Nov 11 '13 at 13:44
  • $\begingroup$ @AnonymousCoward But there is no need to, as $\overline {A_i}=A_i\cup A_i'$ and $A_i\subseteq \mathbb R\setminus\mathbb Q$, so $A_i'\cap\mathbb Q=\emptyset$ is equivalent to $\overline{A_i}\cap\mathbb Q=\emptyset$. $\endgroup$ – Camilo Arosemena-Serrato Nov 11 '13 at 14:03
  • $\begingroup$ @AnonymousCoward, have you checked my edit? $\endgroup$ – Camilo Arosemena-Serrato Nov 13 '13 at 23:54

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