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I think $(0,2)\cup(2,3)$ is dense in [0,3] since the only problem point would be 2

but $B(2,r)\cup [0,3]$ is non-empty for all $r>0$

is this ok?

Joe
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jiboom
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  • That union should be an intersection, but otherwise yes that's the right idea; although you also need to do the same argument for the endpoints $0$ and $3$. – Dave May 13 '18 at 22:26
  • I think that the union in the second line should be an "intersection" (\cap instead of \cup). – John Hughes May 13 '18 at 22:26
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    Right, but why is $2$ any more of a problem point that $0$ or $3$? If the question were "is $(0,2)$ dense in $[0,2]$?" would you say "no worries with $0,$ the only problem point is $2$"? – bof May 13 '18 at 22:30
  • @bof: my thinking was that my set is basically the same without the number 2 in the middle or the end points. so was worried there might be some interval around 2 where the intersection maybe empty as 2 was not in my set. . the endpoints I figured didn't need checking as any ball around a point near 0 or 3 would clearly intersect with [0,3] – jiboom May 14 '18 at 12:18
  • jiboom, it is equally clear that $B(a,r)$ intersects $(0,2)\cup(2,3)$ for all $a = 0,2,3$. You considering one thing clear and the other not is a bit strange. – Ennar May 14 '18 at 15:00

1 Answers1

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Hint: What is the closure of $(0,2)\cup (2,3)$ in $[0,3]$?

Ennar
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