A set under multiplication by non-negative real numbers in Banach space

Question

Let $X$ be a Banach space and $S$ a closed subset of $X$. Is the set $\tilde S=\{rs;\ s\in S,\ r\geq 0\}$ also closed?

Answer 1

No. Take $S=(x=1)\subset\mathbb{R}^2$, for example

Answer 2

This already fails for $X = \mathbb R^2$: $$S = \{ (x,y) \in \mathbb R^2 \mid x,y \ge 0 \text{ and } x \, y = 1\}.$$

However, the result is true if $S$ is assumed to be compact.