TITLE: Prove A is open if and only if w+A is open, A is closed if and only if w+A is closed.
QUESTION [7 upvotes]: Let A be a subset of $\Bbb R^n$ and let $\mathbf w$ be a point in $\Bbb R^n$. The translate of A by $\mathbf w$ is denoted $\mathbf w$ + A and is defined by $$\mathbf w+A \equiv \{\mathbf w + \mathbf u\mid \mathbf u\text{ in A}\}.$$ $\mathbf a.$ Show that A is open if and only if $\mathbf w$ + A is open. $\mathbf b.$ Show that A is closed if and only if $\mathbf w$ + A is closed.  For part $\mathbf a.$ I know that if A is open then $\mathbf u$ is an interior point of A. Is it correct to say: $$ \therefore  \mathbf w + \mathbf u \text{ must be an interior point of } \mathbf w + A \implies \mathbf w + A \text{ is open.}$$  and I'm not sure how to get started on $\mathbf b.$

REPLY [4 votes]: For part (a), it suffices to show that $B_r(w+a) = w + B_r(a)$. But this is easy since
$$x\in B_r(w+a) \iff ||x-(w+a)||<r$$
$$\iff||(x-w)-a||<r$$
$$\iff x-w\in B_r(a)$$
$$\iff x\in w+B_r(a)$$
For part (b), just note that any set $S$ is closed iff $S^c$ is open. So, 
$$A\text{ closed } \iff A^c \text{ open }$$
$$\iff w+A^c \text{ open }$$
$$\iff (w+A)^c \text{ open }$$
$$\iff w+A\text{ closed } $$