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Let $A$ and $B$ be sets such that $A\subset \overline B$ and $B \subset \overline A$.

  1. Isn't that an equivalent statement to saying that $A$ is dense in the closure of $B$ and that $B$ is dense in the closure of $A$? And

  2. What is the technical name for a set $A$ being dense in the closure of $B$ and vice versa, as with $\mathbb{Q}$ and the irrationals?

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up vote 3 down vote accepted
  1. Yes. $A\subset \overline{B}$ means that $\overline{A}\subset \overline{B}$ because $\overline{B}$ is closed and any closed set containing $A$ contains $\overline{A}$. Likewise $\overline{A}\supset \overline{B}$. So the closures are equal, hence each set is dense in the other's closure.

I don't know of any name for this phenomenon.

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$\newcommand{\cl}{\operatorname{cl}}$The answer to your first question is yes.

If $A\subseteq\cl B$ and $B\subseteq\cl A$, then $A\subseteq\cl B\subseteq\cl\cl A=\cl A$, and therefore $\cl A=\cl B$; thus, $A$ is dense in $\cl B$, and $B$ is dense in $\cl A$. Conversely, if $A$ is dense in $\cl B$ and $B$ is dense in $\cl A$, then $\cl A\supseteq\cl B\supseteq\cl A$, so $\cl A=\cl B$, and therefore $A\subseteq\cl B$ and $B\subseteq\cl A$.

I don’t know of any term describing this situation; one simply says that $\cl A=\cl B$.

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