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We define an embedding of the set of prim numbers into the Cantor set as follows:

First we recall that the cantor set $\mathcal{C}$ is homeomorphic to $(\mathbb{Z}/10\mathbb{Z})^\omega $ since the latter is a compact metrizable space without any isolated point. So according to topological characterization of the Cantor set the classical Cantor set is homeomorphic to $(\mathbb{Z}/10\mathbb{Z})^\omega $.

The space of prime numbers is denoted by $\mathcal{Prime}$.

We define the embedding $\mathcal{E}:\mathcal{P}\to (\mathbb{Z}/10\mathbb{Z})^\omega $ as follows:

$$\mathcal{E}(p)=(a_1,a_2,\ldots,a_n,\ldots)$$

where the decimal expansion of $\sqrt{p}=b_nb_{n-1}\ldots b_{1}b_0/a_1a_2\ldots a_n\ldots$

So in this way we may consider the space of prime numbers $\mathcal{Prime}$ as a subspace of the Cantor set $\mathcal{C}$.

Is $\mathcal{Prime}$ a compact set? Is it an open subset of $\mathcal{C}$?

What would be a number theoretical interpretations for these topological questions?

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    $\begingroup$ It may not be that easy to get much out of this doing this in base 10. I have a tiny suspicion that if there is anything more interesting here that base 3 would be the one to use. $\endgroup$
    – JoshuaZ
    Commented Nov 4, 2021 at 21:12
  • $\begingroup$ Number-theretical conclusions will not follow from base 10 expansions. $\endgroup$
    – Gerald Edgar
    Commented Nov 4, 2021 at 21:34

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Your set is a countable dense subset of $(\mathbb{Z}/10\mathbb{Z})^\omega$; cf. Lucia's response here. Hence it is neither open, nor compact.

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    $\begingroup$ Countable sets can sometimes be compact. But I suspect this set is dense, and therefore not compact. $\endgroup$
    – Gerald Edgar
    Commented Nov 4, 2021 at 21:33
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    $\begingroup$ @GeraldEdgar This one is not closed though (since it's a proper dense subset) $\endgroup$
    – Wojowu
    Commented Nov 4, 2021 at 21:34
  • $\begingroup$ @GeraldEdgar I think he means it can not be an open set since it is countable. That is true so when I was giving the question i did not pay attention to this fact $\endgroup$
    – Ali Taghavi
    Commented Nov 4, 2021 at 22:13
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    $\begingroup$ @GeraldEdgar: I said in my post that it was dense, and I gave as a reference Lucia's answer to another MO post. So it cannot be closed (equivalently it cannot be compact). $\endgroup$
    – GH from MO
    Commented Nov 4, 2021 at 22:33
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    $\begingroup$ I thank you for your very helpful answer. $\endgroup$
    – Ali Taghavi
    Commented Nov 10, 2021 at 2:34

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