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I've seen here that the union of two nowhere dense sets is again nowhere dense. Here I ask something milder:

Is the union of two non-dense sets still non-dense?

I would say it is true, but I don't really know how to prove it. Some hint?

Servaes
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Joe
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    Could you give a definition of "non-dense"? If "non-dense" simply means "not dense", then trivial examples show this to be false in a maximal way (e.g. the real line is can be the union of two rays). – Dave L. Renfro Aug 23 '21 at 16:45
  • Why would you say it is true? Can't you include that in your question? – Randall Aug 23 '21 at 16:46
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    From how to ask a good question: Your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title. – jjagmath Aug 23 '21 at 16:49

1 Answers1

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Consider the topological space $X=\{0,1\}$ with the discrete topology.

If you prefer a more 'natural' example, consider $\Bbb{R}$ with its usual topology, and the two subsets $(-\infty,0]$ and $[0,\infty)$. I'm sure you can tell from here that it fails spectacularly.

Servaes
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  • Oh, I see. Thanks. But then, what kind of conditions can we ask for the statement to be true? E.g. If the space is $\Bbb R^N$ with euclidean topology, and we have two sequences ${a_n}_n$ and ${b_n}_n$, none of which dense, can we argue their union is not dense? – Joe Aug 23 '21 at 16:49
  • Of course not; for example $\Bbb{Q}$ is dense in $\Bbb{R}$ and countable. Honestly this is so far from anything true that it barely makes sense to me to try to 'cut it down' to something true. – Servaes Aug 23 '21 at 16:50
  • $\Bbb Q$ is of course dense in $\Bbb R$, but can we split it into two sets, none of which dense? – Joe Aug 23 '21 at 16:52
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    Yes, by the exact same method as in this answer. – Servaes Aug 23 '21 at 16:52
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    Everything right. Thanks – Joe Aug 23 '21 at 16:53
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    +1 for your patience with the OP – hardmath Aug 23 '21 at 17:21
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    You are right, that was a stupid question. And yes, thanks for the patience. But sometimes one is just (really) tired. I guess it happens to everyone – Joe Aug 23 '21 at 18:20