Is the Translation of Open and Closed Sets in R^n Also Open or Closed?

[Appa]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

I tried to solve this componentwise:
$$\vec{u}$$ = {p_i(u_i)}, 1<=i<=n, so that $$\vec{u}$$ + $$\vec{w}$$ = {p_i(u_i) +p_i(u_i)}
But I'm not all that sure whether I'm on the right track..!

 

[Office_Shredder]

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