Let $(X,\rho)$ be a metric space and $A\subset X$. Prove that $\partial A = \overline A \cap \overline{X \setminus A} $. I have no idea how to prove that. Please help.
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How is teh boundary defined? As the closure less the interior? – ncmathsadist May 28 '13 at 23:58
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2Not every question including the word "set" is a question in set theory. – Asaf Karagila May 28 '13 at 23:58
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1What definition of $\partial A$ are you using? Have you solved other problems where you have to show that two sets are equal? – Jonas Meyer May 28 '13 at 23:59
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C'est vrai, Asaf. – ncmathsadist May 28 '13 at 23:59
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My definition is that $\partial A$ is the set of all points all of whose neighborhoods meet both $A$ and $A^c$. – ncmathsadist May 29 '13 at 00:00
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Yes, I have boundary defined as closure less the interior. – user74200 May 29 '13 at 00:03
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@user74200 I hope my answer is helpful. It is hard to give an optimal answer when I don't know what you've proven before, e.g. if you know that $\partial A = \partial (X \setminus A)$. Please comment my answer if anything is unclear or if you have any further difficulty. – Caleb Stanford May 29 '13 at 00:37
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Or you can do that by the definition of the boundary of a set – homomathematicus May 30 '13 at 16:41
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To show two sets are equal, first assume we have an element of the first set, and show it is in the second set. Then, assume we have an element of the second set, and show it is in the first.
Proof outline:
- Let $x \in \partial A$ arbitrary. Show that $x \in \overline{A}$ (this should be easy from your definition). Then, show that $x \in \overline{X \setminus A}$ using the fact that $\partial A = \partial (X \setminus A)$.
- Let $y \in \overline{A} \cap \overline{X \setminus A}$ arbitrary. Show that $y \in \partial A$ by showing that $y \in \overline{A}$ but $y \not \in \text{Int } A$.
Caleb Stanford
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1"using the fact that $\partial A=\partial(X\setminus A)$": Also proving that fact, of course:) – Jonas Meyer May 29 '13 at 00:40
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@JonasMeyer certainly :) See also my comment on the question itself. – Caleb Stanford May 29 '13 at 00:45