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A topological ring is a ring $R$ which is also a Hausdorff space such that both the addition and the multiplication are continuous as maps.

$F$ is a topological field, if $F$ is a topological ring, and the inversion operation is continuous, when restricted to $F\backslash\{0\}$.

If $F$ is a connected field, then $F$ must be path-connected?

David Chan
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  • I never met any connected field other than $\mathbb{R}$ and $\mathbb{C}$ (and these are path-connected). What are your thoughts about? What have you tried so far? – Crostul Nov 11 '15 at 10:02
  • @Crostul $\mathbb{Z}_p$ – David Chan Nov 12 '15 at 03:50
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    $p$-adic integers are totally disconnected! – Crostul Nov 12 '15 at 09:02
  • @Crostul Any field endowed with the trivial topology is a connected topological field. Such a field is also path-connected (and utterly uninteresting). That said, I can't think of non-trivial, non-$\mathbb{R}$ and non-$\mathbb{C}$ examples either. – Pierre-Guy Plamondon Nov 13 '15 at 15:08
  • @Pierre-Guy Plamondon In general, a topological group/ring/field is Hausdorff, here as it well. – David Chan Nov 14 '15 at 11:30
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    I don't think topological rings (or groups, or fields) are defined to be Hausdorff in general (Wikipedia helpfully gives the example of rings with the $I$-adic topology, which are not always Hausdorff). – Pierre-Guy Plamondon Nov 14 '15 at 11:41
  • ..In the def'n of a topological group $G$ it is required that $f : G\times G\to G$ is continuous, where $f(x,y)=x y$ , which is not the same as requiring that $f_x(y)=x y$ and $g_x(y)=y x$ are continuous for each $x$. I assume the same kind of def'n holds for top'l rings and fields for both "+" and "x". – DanielWainfleet Nov 14 '15 at 18:35
  • @David Chan . A $T_0$ top'l group is $T_2$. See Hewitt & Ross,Harmonic Analysis,Vol.I. – DanielWainfleet Nov 14 '15 at 18:39

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A theorem of Pontryagin asserts that $\mathbb{R}$ and $\mathbb{C}$ are the only fields which are both connected and locally compact Hausdorff (LCH), and maybe we need second-countability as well. More generally, there's a known classification of LCH fields that aren't discrete.

Beyond the LCH case it's hard to even get any control on the underlying abelian group. Apparently there are connected topological fields of arbitrary characteristic, though.

Community
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Qiaochu Yuan
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  • On page 134 of Wieslaw's "On topological fields", there are references for the construction of connected (non-locally compact) topological fields of arbitrary characteristic, e.g. this one, see prop. p. 183. The references p. 47 here might also be interesting. – Watson Nov 27 '19 at 20:22