# Open and closed sets in R^n

1. Feb 15, 2009

### Appa

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

The translate of A by $$\vec{w}$$ is defined by
A + $$\vec{w}$$ := {$$\vec{w}$$ + $$\vec{u}$$ | $$\vec{u}$$ in A}

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

2. Feb 15, 2009

### Office_Shredder

Staff Emeritus
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