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Open and closed sets in R^n

  1. Feb 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Let A be a subset of Rn and let [tex]\vec{w}[/tex] be a point in Rn. Show that A is open if and only if A + [tex]\vec{w}[/tex] is open.
    Show that A is closed if and only if A + [tex]\vec{w}[/tex] is closed.
    2. Relevant equations

    The translate of A by [tex]\vec{w}[/tex] is defined by
    A + [tex]\vec{w}[/tex] := {[tex]\vec{w}[/tex] + [tex]\vec{u}[/tex] | [tex]\vec{u}[/tex] in A}

    3. The attempt at a solution
    I tried to solve this componentwise:
    [tex]\vec{u}[/tex] = {pi(ui)}, 1<=i<=n, so that [tex]\vec{u}[/tex] + [tex]\vec{w}[/tex] = {pi(ui) +pi(ui)}
    But I'm not all that sure whether I'm on the right track..!
  2. jcsd
  3. Feb 15, 2009 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    You can just brute force the definition of open: If you have a point u in A, you have a small ball around it contained in A, and that ball will be translated into A+w. You should be able to see how that gives you that A+w is open
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