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Let $p$ be a prime and ${\mathbb C}_p$ be the completion of the algebraic closure $\overline{{\mathbb Q}_p}$. This field is isomorphic to $\mathbb C$. Both fields come with natural absolute values but have very different topologies. There are no continuous field isomorphisms, but that does not exclude Borel-measurable ones (measurable in both directions, that is). Do those exist?

A related question is the question for measurable automorphisms of $\mathbb C$. Are there more than the complex conjugation?

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    $\begingroup$ Concerning the last question: $\mathbb{C}$ is Polish hence by Banachs theorem every (Baire/Borel)-measurable homomorphism is automatically continuous. $\endgroup$
    – Dieter Kadelka
    Commented Aug 8, 2021 at 9:59
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    $\begingroup$ It seems that the topology of $\mathbb{C}_p$ is Polish too (I have not proved it). Look in mathoverflow.net/questions/238809/… (third paragraph). Thus the Borel isomorphisms are continuous too in this case. $\endgroup$
    – Dieter Kadelka
    Commented Aug 8, 2021 at 10:50

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$\mathbb C$ and $\mathbb C_p$ are separable complete metric spaces (Polish spaces). They are Polish groups under addition. A Borel-measurable homomorphism between Polish groups is continuous. (Sometimes called "automatic continuity".) So any such Borel isomorphism must be a homeomorphism. But $\mathbb C$ is connected while $\mathbb C_p$ is totally disconnected, so they are not homeomorphic. We did not require field isomorphism, only group isomorphism for addition.

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  • $\begingroup$ Next, you could try to determine whether $\mathbb C_2$ and $\mathbb C_3$ are connected by a Borel isomorphism. $\endgroup$
    – Gerald Edgar
    Commented Aug 8, 2021 at 13:22

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