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A and A complement have the same boundary

  • Thread starter Ted123
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  • #1
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How would you show that if [itex]X[/itex] is a non-empty set and [itex]A\subseteq X[/itex] then [itex]A[/itex] and [itex]A^c[/itex] have the same boundary?

The definition is [itex]x\in \partial A \iff [/itex] there exists [itex]r>0[/itex] such that the open ball [itex]B(x,r)[/itex] intersects both [itex]A[/itex] and [itex]A^c[/itex]

but this is precisely the statement that [itex]x\in \partial A^c[/itex]!
 

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  • #2
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How would you show that if [itex]X[/itex] is a non-empty set and [itex]A\subseteq X[/itex] then [itex]A[/itex] and [itex]A^c[/itex] have the same boundary?

The definition is [itex]x\in \partial A \iff [/itex] there exists [itex]r>0[/itex] such that the open ball [itex]B(x,r)[/itex] intersects both [itex]A[/itex] and [itex]A^c[/itex]

but this is precisely the statement that [itex]x\in \partial A^c[/itex]!
Indeed, so the proof is trivial.
 

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