Let $W_2\in\Bbb Z_2$ be any finite binary word as an element of the 2-adic integers. Let the relation For all $W:$ $\overline{0}1W_2\sim\overline{1}0W_2$ be a quotient on $\Bbb Z_2$.
When we read the number backwards, this equivalence relation is the same one which sets equivalent binary representations of real numbers the same, e.g. $0.0\overline1=0.1\overline0$. So it defines a quotient map on $\Bbb Z_2$ which gives me the real numbers in the unit interval, at least as a set, when read in reverse.
I'm trying to work out "how homeomorphic" the result $\Bbb Z_2/{\sim}$ is to the real interval. On a prima facie basis it's very similar because it contains all the same elements and they will be close together if they differ by a small power of $2$ (if read as real numbers) or a large power of two if read as p-adic numbers.
But I can see a subtle difference insofar as the distance in the p-adic topology is only the first digit of difference whereas in the real topology, it is the sum of that plus all subsequent digits of difference. I guess this is the essence of the padics being totally disconnected, while $\Bbb R$ is connected. So my tentative (incompetent) topology skills say the map from $\Bbb Z_2/{\sim}\to\Bbb R$ is not a homeomorphism.
But the motivation I need to satisfy, is that I'd like to understand when a function which is continuous in $\Bbb Z_2/{\sim}$ is continuous in $\Bbb R$. It looks to me like any arbitrary function which is continuous in $\Bbb Z/{\sim}$ will also be continuous in $\Bbb R$ despite the map not being a homeomorphism. It looks to me like disconnectedness is only the property that doesn't push through from one to the other, but continuity is preserved. Is this intuit right or wrong?