Let $K$ be a topological field. If $K$ is a connected Hausdorff space, and is algebraically closed, is it true that $K$ is isomorphic to $\mathbb{C}$ ?
(I have deleted my question on MathStackExchange)
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4$\begingroup$ Apparently Dieudonné has constructed proper dense subfields of $\mathbb{C}$ which are connected and algebraically isomorphic to $\mathbb{C}$. I don't know if this answers the question since the meaning of "isomorphic" is unclear. See chapter X of Wieslaw's book. In the same chapter there is this result of Bergman and Waterman (Theorem 7): "Every discrete field can be embedded in an arcwise connected topological field". But of course algebraic closedness is not mentioned. $\endgroup$– Laurent Moret-BaillyCommented Nov 10, 2021 at 20:00
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$\begingroup$ This answers the question. Sorry, I was not clear. By "isomorphic", I mean algebraically and topologically isomorphic. $\endgroup$– marco2013Commented Nov 10, 2021 at 20:19
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2$\begingroup$ Side remark: Even without connectedness assumption: every locally compact topological field that is algebraically closed is isomorphic to $\mathbf{C}$ as topological field. $\endgroup$– YCorCommented Nov 10, 2021 at 21:33
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