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Open and closed sets in R^n

  • Thread starter Appa
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  • #1
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Homework Statement


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.

Homework 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}

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..!
 

Answers and Replies

  • #2
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
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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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