Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

2 Answers 2

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.

share|improve this answer

$\newcommand{\cl}{\operatorname{cl}}$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 $\cl A=\cl B$.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.