# Is the reals the smallest connected ordered topological ring?

The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to define the reals is that it is (Dedekind- or Cauchy-) complete ordered field.

Consider the real numbers among other ordered topological rings. I am wondering if the real numbers is the smallest connected ordered topological ring? Here "smallest" means that it embeds into every other connected ordered topological ring. If this is not true, do you have some minimal collection of additional adjectives (advoiding the word "complete") that can characterize the real numbers among ordered topological rings? "Minimal" means that if you drop an adjective then you there are other ordered topological rings that also fulfil your definition.

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Since the reals are characterized by being the only complete totally ordered field, any other criterion you give will have to have something with the feeling of completeness to it- I don't think you can get something from nothing, especially if the "something" is "real numbers". –  Dylan Wilson Feb 9 '12 at 6:56
In lights of the comments above, I would suggest editing the title. Perhaps to something like "characterizing $\mathbb R$ as a topological ring"... –  Gjergji Zaimi Feb 9 '12 at 8:47
argh, i forgot to add "connected". I meant to ask if the reals was the smallest connected ordered topological ring. –  Colin Tan Feb 9 '12 at 13:26
this is discussed in the comments to Gjerigji's answer. –  Colin Tan Feb 9 '12 at 13:28
Must an ordered ring be an integral domain? –  Gerald Edgar Feb 9 '12 at 14:59
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## 1 Answer

The following characterization of $\mathbb R$ and $\mathbb C$ among topological rings is due to Pontryagin and seems to be in the spirit of your question:

Theorem: If $F$ is a field with a Hausdorff ring topology which is locally compact and connected then $F$ is isomorphic as a topological field to either $\mathbb R$ or $\mathbb C$ with their usual topologies. (If you substitute "field" with "division ring" then you must add the quaternions $\mathbb H$ to the list.)

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But, as I said, you have to have some complete-ish feeling to your conditions... Locally compact Hausdorff is a nice one. (Also, en.wikipedia.org/wiki/…) –  Dylan Wilson Feb 9 '12 at 7:21
I agree. The surprise here, for me, is the lack of other properties like "ordered" etc. from the statement. Locally compact Hausdorff connected seemslike too little at first sight :) –  Gjergji Zaimi Feb 9 '12 at 7:35
Are there easy counterexamples if the "locally compact" condition is dropped? Now that I think about it, I really don't know a lot of connected fields... –  Qiaochu Yuan Feb 9 '12 at 7:50
I can't think of any simple examples, though I read that any field with the discrete topology embeds in a connected field. It seems that it is not known whether any field embeds in a connected field... –  Gjergji Zaimi Feb 9 '12 at 8:20
Waterman, Alan G., Bergman, George M., "Connected fields of arbitrary characteristic." J. Math. Kyoto Univ. 5 (1966) 177–184. Here we have the proof that an arbitrary discrete field may be imbedded in a connected field. –  Gerald Edgar Feb 9 '12 at 14:57
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