- #1
Appa
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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..!