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So far in my class, we've defined the real numbers as a collection of all Dedekind cuts. Additionally, we've proven that any nonempty subset of the real numbers that is bounded above has a supremum. Using this, I am asked to prove that the real numbers are connected (i.e., they can't be assembled from the union of two nonempty, disjoint, and open sets). We have already proven that a continuum (we have not yet defined the real numbers as the only continuum) is connected if all bounded and nonempty subsets have supremum. However, the proof is very long and complicated and has nothing to do with Dedekind cuts; while I could just recycle this old proof, I'm sure there is probably a simpler proof I am missing that can use the fact that the real numbers are made of Dedekind cuts. Any proofs or hints would be greatly appreciated.

By the way, if anything I've said does not make sense or if you'd like me to provide more information about anything, just let me know.

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  • $\begingroup$ The word you're looking for is connected. This is much better than "continuous" in this context - "continuous" really should apply to functions and things like that. The proof is not that long, and Dedekind cuts correspond excellently well with suprema. $\endgroup$
    – FShrike
    Nov 18 at 23:08
  • $\begingroup$ Can you prove the law of trichotomy for the reals? $\endgroup$
    – CyclotomicField
    Nov 18 at 23:34
  • $\begingroup$ So you also defined a topology on $\mathbb R$? And what is the definition of "continuum"? $\endgroup$
    – Paul Frost
    Nov 20 at 18:20

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