I want to ask about the homeomorphism between $\mathbb{Q}$, $\mathbb{Q}_{>0}$: the rationals greater than $0$, and $\mathbb{Q}_{\geqslant 0}$: the rationals $\geqslant 0$?
For $\mathbb{Q}$ and $\mathbb{Q}_{>0}$, I can use the back and forth map to have an order-isomorphism, and as a result, a homeomorphism. Is there a direct formula for the map?
It's very surprising that $\mathbb{Q}_{\geqslant 0}$ is homeomorphic to $\mathbb{Q}_{>0}$ and $\mathbb{Q}$, I don't know how to show they are homeomorphic.
Could anyone please help me with the homeomorphisms between them?