If $A,B\subset\mathbb{R}$ are closed, is $AB$ closed?

Question

From this question Sum of two closed sets in $\mathbb R$ is closed? We see that the sum of $2$ closed subsets of $\mathbb{R}$ is not necessarily closed. I wondered if a similar result holds for their product.

Given $A,B\subset \mathbb{R}$, we define $AB:=\{ab: a\in A, b\in B\}$.

I have the following observation:

If $A$ is compact, $0\notin A$, and $B$ is closed, then $AB$ is closed.

Suppose $c$ is a limit point of $AB$. Then there exists $(c_n)\subset AB$ such that $(c_n) \to c\in \mathbb{R}$. Let $c_n=a_nb_n$, where $(a_n)\subset A$ and $(b_n)\subset B$. Since $A$ is compact, then there exists a subsequence $(a_{n_k})$ of $(a_n)$ such that $(a_{n_k})\to a$ for some $a\in A$. Note that $b_{n_k}=\frac{c_{n_k}}{a_{n_k}}$. So $(b_{n_k})\to \frac{c}{a}$. Since $B$ is closed, we have $\frac{c}{a}\in B$. It follows that $c\in AB$. Therefore $AB$ is closed.

I tried to find a counterexample to my claim at the top, but I couldn't come up with one.

A simple transformation can be used to get a counter-example using the case of sum of two closed sets. — Kavi Rama Murthy, Oct 01 '24 at 08:09
@geetha290krm I think the example below is better because it gives a good reason why $0\notin A$ in OP's theorem is important — Jakobian, Oct 01 '24 at 08:48

score 7 · Accepted Answer · answered Oct 01 '24 at 08:08

7

Let $A=\mathbb Z$ and $B=\{0\}\cup \{{1\over n}:n\in \mathbb N\}$. In this case ,$AB=\mathbb Q$, which is not a closed subset of $\mathbb R$.

answered Oct 01 '24 at 08:08

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If $A,B\subset\mathbb{R}$ are closed, is $AB$ closed?

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