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The following questions occurred to me while browsing this site and looking at Exercise 20 here.

Question 1. Let $n>1$. Does there exist a countable dense subset $A\subset\mathbb{R}^n$ for which the complement $\mathbb{R}^n\setminus A$ is connecteddisconnected?

EDIT. I deleted an erroneous paragraph. Let me add two more questions, the first one being Gerald Edgar's comment here, the second one correcting the erroneous paragraph.

Question 2. Let $n>1$. Is it true that for any subset $A\subset\mathbb{R}^n$ of Hausdorff dimension less than $n-1$ the complement $\mathbb{R}^n\setminus A$ is connected?

It seems that Joel's argument answers this in the affirmative as well.

Question 3. Let $n>1$. Are there two countable dense subsets $A,B\subset\mathbb{R}^n$ whose complements are not homeomorphic?
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The following questions occurred to me while browsing this site and looking at Exercise 20 here. Let $n>1$:

Question 1. Let $n>1$. Does there exist a countable dense subset $A\subset\mathbb{R}^n$ for which the complement $\mathbb{R}^n\setminus A$ is connected?

EDIT. I deleted an erroneous paragraph. Let me add two more questions, the first one being Gerald Edgar's comment here, the second one correcting the erroneous paragraph.

Question 2. Let $n>1$. Is it true that for any subset $A\subset\mathbb{R}^n$ of Hausdorff dimension less than $n-1$ the complement $\mathbb{R}^n\setminus A$ is connected?

It seems that Joel's argument answers this in the affirmative as well.

Question 3. Let $n>1$. Are there two countable dense subsets $A,B\subset\mathbb{R}^n$ whose complements are not homeomorphic?

According to Exercise 20 here such subsets exist, but I cannot see an example right away.
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Connectedness of the complement of a countable dense subsetsmall subsets (extended question)

The following question questions occurred to me while browsing this site:. Let $n>1$:

Question 1. Does there exist a countable dense subset $A\subset\mathbb{R}^n$ for which the complement $\mathbb{R}^n\setminus A$ is connected?

For sets of

EDIT. I deleted an erroneous paragraph. Let me add two more questions, the form first one being Gerald Edgar's comment here, the second one correcting the erroneous paragraph.

Question 2. Is it true that for any subset $A=B^n$ (i.e. A\subset\mathbb{R}^n$ of Hausdorff dimension less than $B\subset\mathbb{R}$ is countable and dense) n-1$ the complement $\mathbb{R}^n\setminus A$ is homeomorphic to connected?

It seems that Joel's argument answers this in the Baire space, hence it is totally disconnectedaffirmative as well.On the other hand, not all complements

Question 3. Are there two countable dense subsets $\mathbb{R}^n\setminus A$ above A,B\subset\mathbb{R}^n$ whose complements are not homeomorphicto the Baire space, at least according ?

According to Exercise 20 here whose solution such subsets exist, but I would also appreciatecannot see an example right away.
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