Let $a,b\in (0,1)\subset\mathbb{R}$,
$a=0,a_1 a_2 ...$; $\;b=0,b_1 b_2 ...$
Why is
$\pi:\mathbb{R}²\rightarrow\mathbb{R}: (a,b)\mapsto 0,a_1 b_1 a_2 b_2...$
not bijective, if the constraint $\forall N\in\mathbb{N}\exists i>N:a_i\neq9$ is applied?
Let $a,b\in (0,1)\subset\mathbb{R}$,
$a=0,a_1 a_2 ...$; $\;b=0,b_1 b_2 ...$
Why is
$\pi:\mathbb{R}²\rightarrow\mathbb{R}: (a,b)\mapsto 0,a_1 b_1 a_2 b_2...$
not bijective, if the constraint $\forall N\in\mathbb{N}\exists i>N:a_i\neq9$ is applied?
It is true that $\mathbb R$ and $\mathbb R^2$ are isomorphic as sets, as groups, or as vector spaces over $\mathbb Q$.
They are not isomorphic as topological spaces, as rings, or as vector spaces over $\mathbb R$ though.