I have two questions about homeomorphism and isometry.
First, I know that $\mathbb{R}$ and $\mathbb{Q}$ are not homeomorphic, but the proof that I know use the completeness of $\mathbb{R}$. My question is: is there another proof with a different argument? The cardinality is another aspect that differentiates $\mathbb{R}$ from $\mathbb{Q}$ so, should a homeomorphism preserves the cardinality? If yes, just a reference for some proof is enough.
Second, the square and circle (viewed as subsets of $\mathbb{R}^2$) are homeomorphic (we can use the poolar coodinates to get a homeomorphism). But are they isometric (with the standard metric of $\mathbb{R}^2$)?
A square is never isometric to a circle.
For any point $P$ in a (bounded) metric spaces $(M,d)$, we can consider the "furthest distance from $P$", i.e. the quantity $f(P)=\sup_{Q\in M}d(P,Q)$.
For a circle, the value of $f(P)$ is the same for all $P$, while for a square, $f(P)$ is not always the same.
As to the proof that $\Bbb Q$ is not homeomorphic to $\Bbb R$, of course you can use the cardinality argument. Besides, there are other possibilities: for example, $\Bbb Q$ is disconnected, as it can be written as a disjoint union of two open sets $\{x:x^2<2\}$ and $\{x:x^2>2\}$, while $\Bbb R$ is connected.