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I have the feeling I'm missing something obvious, but here it goes...

I'm trying to prove that for a subset $A$ of a topological space $X$, $\overline{\overline{A}}=\overline{A}$. The inclusion $\overline{\overline{A}} \subseteq \overline{A}$ I can do, but I'm not seeing the other direction.

Say we let $x \in \overline{A}$. Then every open set $O$ containing $x$ contains a point of $A$. Now if $x \in \overline{\overline{A}}$, then every open set containing $x$ contains a point of $\overline{A}$ distinct from $x$. My problem is: couldn't $\{x,a\}$ potentially be an open set containing $x$ and containing a point of $A$, but containing no other point in $\overline{A}$?

(Also, does anyone know a trick to make \bar{\bar{A}} look right? The second bar is skewed to the left and doesn't look very good.)

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You can try and prove two separate things that will let you conclude that: $(1)$ $\operatorname{cl} A$ is closed. $(2)$ If $A$ is closed, then $\operatorname{cl} A = A$. The fact that ${\operatorname{cl} ({\operatorname{cl} A})}=\operatorname{cl} A$ then follows at once. –  Pedro Tamaroff Jul 20 '12 at 2:10
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3 Answers 3

up vote 4 down vote accepted

How do you define the closure of a set $A$? If it's not already your definition, it might be a useful thing to prove that the closure of a set is precisely the intersection of all closed sets containing it.

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Which is, the smallest closed set containing it... –  Temitope.A Jul 20 '12 at 9:43
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The condition you want to check is \[ x \in \bar A \quad \Leftrightarrow \quad \text{for each open set $U$ containing $x$, $U \cap A \neq \emptyset$} \] This definition implies, among other things, that $A \subset \bar A$. Indeed, with the notation above we always have $x \in U \cap A$. Is it clear why this implies the remaining inclusion in your problem?

If you instead require that $U \cap (A - \{x\}) \neq \emptyset$ then you have defined the set $A'$ of limit points of $A$. We have $\bar A = A \cup A'$. Simple examples such as $A = \{0\}$ inside of $\mathbb R$, for which $\bar A = A$ but $A' = \emptyset$, can be helpful in keeping this straight.

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Most topological properties, by the way, have a lot of equivalent forms. Learning the different definitions of the closure listen on Wikipedia can really save you work — Kris' hint, for example, would give maybe the quickest proof possible of the inclusion that you already did. –  Dylan Moreland Jul 20 '12 at 2:13
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Suppose $x\in\overline{\overline{A}}$. Let $U$ be an open set containing $x$; we want to show that $U\cap A\neq\varnothing$. We know that $U\cap\overline{A}\neq\varnothing$, so there exists $y\in \overline{A}$ such that $y\in U$. But since $U$ is an open set that contains $y$ and $y\in\overline{A}$, then...

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