Closed sets and subtraction of elements.

Question

Let $X$ be a normed vector space, $A,B\subset X$ closed and $M=A-B:=\{x\in X:x=a-b ,\quad a\in A, b \in B\}$.

I am looking for an example where $A,B\quad$ closed $\Longrightarrow\quad M\quad$ closed

is wrong.

Answer 1

Consider $A = \mathbb N_{\ge 2}$ and $B = \{ n + \tfrac1n \mid n\in A \}.$

Notice that $A,B$ are in fact closed. For example, for any sequence $b_k = n_k + \frac{1}{n_k}\in B$ we have $$ |b_k - b_l| \ge |n_k - n_l| - \left| \frac{1}{n_k} - \frac{1}{n_k} \right| \ge |n_k - n_l| - \frac{1}{2}. $$ So, $b_k$ converges if and only if $b_k$ is eventually constant.

Now, $M = A-B$ contains the sequence $m_k = -\frac{1}{1+k}$, which converges to $0\notin M$. That is, $M$ isn't closed