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Mysteriously simple containment questions

  1. May 20, 2013 #1
    Prove that the set i) S[itex]\subseteq[/itex] S(Closure)
    ii) (Sint)int = Sint

    Ok these supposedly simple containment questions irk me every time, how simplistic do I have to unravel the darn definitions:

    Attempts: i) Let x [itex]\in[/itex] S(Closure)
    ==> x[itex]\in[/itex]S or x[itex]\in[/itex][itex]\partial[/itex]S (Boundary of S)
    ==> if x[itex]\in[/itex]S the there exists a B(r,x) [itex]\subseteq[/itex] S
    likewise if x[itex]\in[/itex][itex]\partial[/itex]S then there exists the
    B(r,x)[itex]\cap[/itex]S ≠∅ and B(r,x)[itex]\cap[/itex]Sc≠∅

    ==> S[itex]\subseteq[/itex] S(Closure) I really don't see what else can be done

    ii) isn't this just direct from the definition of the Sint? I mean it's the interior of an interior.
  2. jcsd
  3. May 20, 2013 #2


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    Homework Helper

    i) First try to prove a simpler statement that is analogous to what you are asked to prove. For example, how would you prove that S⊆ S?

    PS. This is all the help I can give here.
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