I think the (set of the) ring of 2-adic integers is represented by:
$$\left\{\sum_{i=0}^\infty 2^ix_i:x_i\in\{0,1\}\right\}$$
And I think the set of real numbers in the interval $[0,1)$ is represented by:
$$\left\{\sum_{i=0}^\infty 2^{-(i+1)}x_i:x_i\in\{0,1\}\right\}$$
Is it correct therefore, that there's a straightforward homeomorphism and order-isomorphism between the two given by reversal of the binary string?
Assuming that all checks out, does the homeomorphism extend to any other structures? e.g. It seems $[0,1)\cong\Bbb R/\Bbb Z$ is a group with addition modulo $1$. But the obvious answer seems to be that addition doesn't correspond because carries go the opposite way.
No, they're not homeomorphic because in $\Bbb R$ we have $0\overline{1}_2=1\overline0_2$ such as $0.\overline1_2=1$ whereas in $\Bbb Z_2$ we have $\overline1_2=-1\neq2^\infty=0$. In fact $[0,1]$ can be realised from $\Bbb Z_2$ by applying the quotient map $\Bbb Z_2\to[0,1]$ that glues together equivalent binary fractions.