I got a good answer to this question over on MathOverflow a while ago. Harald Hanche-Olsen claimed that, if $f, g: D\to \mathbb{R}^+$, then $ f(x) \sim g(x) \implies f(x) \asymp g(x) \qquad \qquad (*) $ holds whenever $D\subseteq\mathbb{C}$ is closed in $\mathbb{C}$, and is false whenever it is not closed.
However, something bugs me about this. Most instances I've seen this, it does hold when $D$ is the strictly positive reals, which I believe is not closed in $\mathbb{C}$.
So this is my question. Does $(*)$ hold when $f, g: \mathbb{R}^+ \to \mathbb{R}^+$? That is, when $f$ and $g$ are positive real-valued and defined on the positive reals?
(Edit: For a detailed explanation of which definitions I am using, see the link. I believe they are standard.)