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Let $U= cup C$ be an open set in $mathbbR^n$. Show there is a countable number of closed balls $beta=B_n=overlineB_r_n(x_n):nin mathbbN$ such that:

a.$U=cup_n B_n^textint$

b. For each $nin mathbbN$ there exists an open set $M$ such that $B_nsubseteq M$.

c. For every point $xin U$, there exists a neighbourhood $V$ of $x$ such that only finitely many elements of the cover $C$ intersect $V$ non trivially.

So I want to use compact exhaustion of the open set $U$.

Fix a compact exhaustion $K_n:ninmathbbN$ of $U$

Then since by definition for every $iinmathbbN$ $K_isubseteq K_i+1^textint$

I can make a compact set $L_n=K_nsetminus K_n-1^textint$ which will satisfy that each $L_i$ is compact as its an intersection of closed sets, which will be closed and is bounded by $K_i$.

From here I think what I want to do is to take balls $B_i(x)$ for each $xin L_i$ and take a radius which will not intersect more then $L_i+1$ or $L_i-1$. A set of all such balls $B_n$ in each $L_i$ will admit a finite subcover because each $L_i$ is compact. And the union of all such finite subcovers of each $L_i$ should still cover $U$ and because $L_n$ is countable this set $cup B_n$ will be countable.

I believe this would satisfy part a. but I don't really see part b and c. I also don't know what a non trivial intersection is.

Here's an answer to part a: This is just the same as, or a slight modification of the proof that was posted earlier.

For any set $U$ the set of balls $$mathcalB=~qin mathbbQ^n, p in mathbbQ_ge 0$$ form a countable cover of $U$.

Let $x$ be an arbitrary point in $U$. Then $x$ is contained in an open ball $B_r_x(x) subseteq U$. Let $q$ be a rational point contained within the ball $B_r_x/3(x)$. Then we can choose a ball $B_p(q) in mathcalB$ with $r_x/3 < p <2r_x/3$. Then $$x in B_p(q) subset B_r_x(x) subseteq U.$$

Hence $U$ is the union of a subset of $mathcalB$.

Take a family $V=V_n:nin Bbb N$ of bounded open sets such that $U=cup V$ and such that $overline V_n subset V_n+1$ for each $n.$ (To get $V,$ see APPENDIX below.)

Let $K_1=overline V_1.$ For $n>1$ let $K_n=bar V_n V_n-1.$ Each $K_n$ is compact.

Let $C_1$ be a finite cover of $K_1$ by open balls, such that $overline cup C_1subset V_2.$ This is possible because $K_1$ is compact, $V_2$ is open, and $K_1subset V_2.$

For $n>1$ let $C_n$ be a finite cover of $K_n$ by open balls, such that $overline cup C_nsubset V_n+1$ and also such that $cup C_n$ is disjoint from $cup_jle n-2(overline cup C_j).$

The last condition above is vacuous when $n=2.$ When $n>2$ it is crucial, and possible because $S_n=cup_jle n-2(overline cup C_j)$ is a subset of $cup_jle n-1V_j=V_n-1,$ and $V_n-1$ is disjoint from $K_n,$ so the closed set $S_n$ and the compact set $K_n$ are disjoint.

Now $C=cup_nin Bbb nC_n$ is a countable family of open balls, and $U=cup C=cup _bin C, int(bar b)=cup_bin C,bar b.$

If $pin U$ then for some $n$ we have $pin V_nsubset overline V_n=cup_jle nK_j$ so for some $jle n$ and some $b in C_j$ we will have $pin bin C_j.$ Now this $b$ is disjoint from $cup C_m$ for every $mge n+2.$ So $b'in C: b'cap bne phi$ is a subset of the finite set $cup_jle n+1C_j.$

APPENDIX. Let $bar 0$ be the origin in $Bbb R^n.$

(i). If $U=Bbb R^n$ let $V_n=B_n(bar 0)$ for each $nin Bbb N.$

(ii). If $Une Bbb R^n,$ let $U^c=Bbb R^n$ $U$ and let $d$ be a metric for the topology on $Bbb R^n,$ and for $pin U$ let $d(p,U^c)=inf d(p,q):qin U^c.$

Now let $V_1=pin Ucap B_1(bar 0): d(p,U^c)>1.$ And for $n>1$ let $V_n= pin Ucap B_n(bar 0): d(p,U^c)>1/n.$ I will leave it to the reader to confirm that $V_n:nin Bbb N$ has the required properties.