(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hi guys, this problem gave me some trouble before, but I'd like to know if I have it worked out now.....

"If S = S[tex]\cup[/tex]BdyS, then S is closed (S[tex]_{compliment}[/tex] is open)

2. Relevant equations

S is equal to it's closure.

3. The attempt at a solution

1. Pick a point p in S[tex]^{compliment}[/tex].

2. For all points q[tex]^{n}[/tex] in S, let [tex]\delta[/tex] = min{|p-q[tex]^{n}[/tex]|}

3. Dfine B(p,[tex]\delta[/tex])

4. For any point in S[tex]^{compliment}[/tex], we can produce [tex]\delta[/tex]>0 such that B(p,[tex]\delta[/tex])[tex]\subset[/tex]S[tex]^{compliment}[/tex]. Therefore, all points in S[tex]^{compliment}[/tex] are interior points; therefore, S[tex]_{compliment}[/tex] is open, and S is closed.

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# Homework Help: Closed set equivalence theorem

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