Except for the trivial case of the partition consisting of the real number $0$ and everything else, is there any other partition of the entire set of real numbers $\mathbb{R}$ into two sets $A$ and $B$ such that $A$ and $B$ are both closed under multiplication?
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3What have you tried? – Thomas Andrews Apr 19 '23 at 15:42
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2There is a pretty basic example... – Thomas Andrews Apr 19 '23 at 15:43
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2Think of the magnitude of the products. – David Mitra Apr 19 '23 at 15:46
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@ThomasAndrews Which one? OP already discarded ${{0}, \mathbb{R}^*}$. – Bruno B Apr 19 '23 at 15:47
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2@BrunoB I'm trying not to answer the question with so little work, but there is an another example. (Well, a couple, but all related.) – Thomas Andrews Apr 19 '23 at 15:57
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I see, thanks for the heads up, and sorry for bothering you. (And yeah, just thought of an example as well) – Bruno B Apr 19 '23 at 15:57
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2Oh, I just thought of the interval $[-1,1]$ and its complement. – user107952 Apr 19 '23 at 17:53
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Are there instances when neither A nor B is an interval? – SpectreDNZ Apr 19 '23 at 18:39
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@SpectreDNZ I feel like that would be an interesting question. Something that I linked earlier (and that I'll link right now) about the positive numbers might serve, but it would be probably advisable to make it another question altogether. Also note that one of the main links provided there has died it seems, at least on my end. The post in question: https://math.stackexchange.com/q/2285600/1104384 – Bruno B Apr 19 '23 at 19:04