# Formally prove the closure and interior of this set

In $\mathbb{R^2}$ with the usual metric, consider $C_n=\{(x,y)\in\mathbb{R^2} : x^2 + y^2 = 1/n^2\}$, that is, the circumference centered at the origin with radius $1/n$. Now consider the union of all those, that is, $C=\cup_{n\in\mathbb{N}}C_n$

In an exercise, I'm asked to calculate the closure, interior and boundary. Now, the intuitive solution to me is that the closure is $C\cup\{(0,0)\}$, the boundary is the same, and the interior is empty. But how to prove this formally?

EDIT: Regarding what I've tried, we firstly know that $C$ is in the closure. Then, do we have anything else? By using the characterization of the closure by sequences, its easy to prove that you can find a sequence that converges to the origin, hence it also is in the closure. Now, what I do is consider a point $p\notin C, p\neq(0,0)$, and try to prove that there are no possible sequences in $C$ that converge to $p$. I've also tried to prove that there exist an open balls centered at that point contained in $X-C$. The problem for me always arises at the time of formalizing those thoughts.

• What did you try? Did you try to check that the conjectured closure is closed? That C is dense in the conjectured closure? Jun 5, 2014 at 0:58
• Well this is what I did: Obviously we know that $C$ is in the closure. Then, do we have anything else? By using the characterization of the closure by sequences, its easy to prove that you can find a sequence that converges to the origin, hence it also is in the closure. Now, what I do is consider a point $p\notin A, p\neq(0,0)$, and try to prove that there are no possible sequences in $A$ that converge to $p$. I've also tried to prove that there exist an open balls centered at that point contained in $X-A$. The problem for me always arises at the time of formalizing those thoughts. Jun 5, 2014 at 1:07
• Try to show that the union of C and the origin is closed (for instance by identifying it with preimage of a closed set under a continuous mapping). Jun 5, 2014 at 1:12

Suppose $p \notin C$ and $p \neq (0,0)$. If $\| p\| > 1$, then let $\delta = (\|p\|-1)/2$ and show that every $x \in B_{\delta}(p)$ is such that $\| x \| > 1$ and hence $B_{\delta}(p) \subset C^c$. Similarly, if $\|p\|<1$, then there is some $n$ such that $1/(n+1)<\|p\|<1/n$; show that if $\delta = \min\{1/n-\|p\|, \|p\| - 1/(n+1)\}/2$ then every $x \in B_{\delta}(p)$ is such that $1/(n+1) < \|x\| < 1/n$ and hence $B_{\delta}(p) \subset C^c$. Don't forget about the triangle and reverse triangle inequalities. From this you can deduce that $(C\cup\{(0,0)\})^c$ is open.