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I am trying to solve the following exercise:

If $U$ is an open subset of $\mathbb{R}^n$, show that there exists an increasing sequence $\{A_k\}^\infty_1$ of compact contented sets such that $U=\bigcup^\infty_{k=1}\ \mathrm{int}\ A_k$.

Hint: Each point of $U$ is contained in some closed ball which lies in $U$. Pick the sequence in such a way that $A_k$ is the union of $k$ closed balls.

It seems "increasing sequence" means that $A_k \subset A_{k+1}$ for all $k$.

Now, I think the hint says that $A_{k+1}$ should be $A_k \cup B_{k+1}$ for some suitable ball $B_{k+1}$, starting with $A_1 := B_1$. But I don't see a way to choose those balls so that eventually all points of $U$ are covered?

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    Do you want $A_k\subset U$? (You can certainly obtain this.)2012-01-05

2 Answers 2

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HINT: Every open set in $\mathbb{R}^n$ is a union of open (or closed) balls whose centres have rational coordinates and whose radii are rational; how many such balls are there?

In case the first statement isn’t obvious, suppose that $B$ is the ball of radius $r$ about a point $x\in\mathbb{R}^n$. If $x$ has all rational coordinates, there’s nothing to be done. Otherwise there is a point $y$ with all coordinates rational inside the ball of radius $r/2$ centred at $x$. Let $d$ be the distance between $x$ and $y$, let $q$ be a rational number such that $d, and let B' be the ball of radius $q$ centred at $y$; then x\in B'\subseteq B.

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    Thanks for this answer! I need further clarification on it though, and asked a question about it here: http://math.stackexchange.com/questions/1922230/how-to-show-that-the-set-of-open-balls-with-rational-centres-and-rational-radii -- would you be able to comment/answer there too?2016-09-11
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To extend Brian's hint:

First cover $U$ as suggested by Brian by "rational" open balls whose closures are contained in $U$ . You may enumerate this covering: $ U=\bigcup_{i=1}^\infty F_i. $ where each $F_i$ is an open ball with $\overline F_i$ contained in $U$.

Now consider the sequence: $A_1=\overline{F_1}\ $, $A_2=\overline{F_1}\cup\overline{ F_2}\ $, $A_3= \overline{F_1}\cup \overline{F_2}\cup\overline{ F_3}\ $, $\ldots$.

Note that: finite unions of compact sets are compact

and

$F_i\subset \text{int} A_i$.