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Real Analysis: Interior, Closure and Boundary

  1. Jan 21, 2010 #1

    michael.wes

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    Gold Member

    1. The problem statement, all variables and given/known data
    Let [tex] W\subset S \subset \mathbb{R}^n.[/tex] Show that the following are equivalent: (i) [tex]W[/tex] is relatively closed in [tex]S[/tex], (ii) [tex]W = \bar{W}\cap S[/tex] and (iii) [tex](\partial W)\cap S \subset W[/tex].

    2. Relevant equations
    The only thing we have to work with is the definitions of open and closed sets, relatively open and relatively closed sets, the result that the complement of an open set is closed, the definition of boundary, interior and closure.


    3. The attempt at a solution
    I have proved that (ii) implies (i). I need help with (i) implies (iii) and (iii) implies (ii).

    For (i) implies (iii), there exists a closed set [tex]C[/tex] such that [tex]W=C\cap S[/tex]. Now try to derive that [tex]\partial W \cap S \subset W[/tex]. Write [tex]\partial(C\cap S)\cap S[/tex], but after this all I am able to do is draw a diagram and get stuck.

    For (iii) implies (ii), we have that [tex]\partial W \cap S \subset W[/tex] and we need to derive [tex]W = (W\cup\partial W) \cap S[/tex], but I can't get started.

    Please help. Thank you!
     
  2. jcsd
  3. Jan 21, 2010 #2
    Try the direction (i) => (ii) => (iii) => (i) instead. You will need the relation [tex]\overline{W} = W^\circ \cup \partial W[/tex].
     
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