Proving that there is no set with a certain given set of limit points

Question

The first question asked of me was to construct a set of points with limit points being the integers. I came up with $$A=\cup_{n=1}^\infty \{(n-1)+1, (n-1)+1/2,(n-1)+1/3+... \} \cup \cup_{n=1}^\infty \{-n-1,-n-1/2,-n-1/3,... \}$$ based on an example my instructor did.

The second question, which I am confused about, is: Prove that there is no set $A$ whose set of limit points is $$\{1,1/2,1/3,1/4,...,1/n,...\}$$ The hint for the second question is to understand why the first question works, but I don't see why on a deep level. Please help me answer this second question and better understand the first.

The second fails because if all the $\frac 1n$ are limit points, then $0$ must be a limit point. — lulu, Sep 04 '21 at 22:57
It's not on the list. If you add $0$ to the list, then you can do it. — lulu, Sep 04 '21 at 23:02

score 1 · Accepted Answer · answered Sep 04 '21 at 23:06

1

Hint: For every set $A$ it holds $A'' \subseteq A'$. Now if there exists a set $A$ with $A' = \{1/n: n \in \mathbb{N}\}$ by the above $0 \in A'' \subseteq A'$, a contradiction.

answered Sep 04 '21 at 23:06

Giorgos Giapitzakis

6,416

Is $A'$ notation for the limit points of $A$? – user404974 Sep 04 '21 at 23:07
@ExponentBOT Yes. – Giorgos Giapitzakis Sep 04 '21 at 23:08
Where can I find a proof of "For every set A it holds A′′⊆A′"? – user404974 Sep 04 '21 at 23:10
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@ExponentBOT I suggest you try and prove it yourself. If you get stuck here is an MSE post about it – Giorgos Giapitzakis Sep 04 '21 at 23:11
Okay, so the idea is that the set $A'$ of limit points (for any set $A$) is always closed, and hence it contains its limit points, i.e. $A'' \subseteq A'$. – user404974 Sep 04 '21 at 23:21
@ExponentBOT Exactly. I don't know how familiar you are with topology but this result holds generally for $T_1$ topological spaces (ie singletons are closed). In the case of metric spaces, it's pretty trivial to prove it just by comparing the sequence characterizations for closed sets and limit points. – Giorgos Giapitzakis Sep 04 '21 at 23:24

Proving that there is no set with a certain given set of limit points

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