# Closed subset of unit ball with peculiar connected components

Let $$n\geq 2$$ and denote by $$B\subset \mathbb{R}^n$$ the closed unit ball.

Does there exist a closed subset $$A\subset B$$ containing $$0\in \mathbb{R}^n$$ with the following properties i,ii,iii?

i) $$\{0\}$$ is a connected component of $$A$$.

ii) $$0\in \overline{A\setminus\{0\}}$$ and

iii) each connected component $$C\neq \{0\}$$ of $$A$$ intersects $$\partial B$$: $$C\cap \partial B\neq \emptyset$$.

A comment: $$A$$ needs to have infinitely many connected components by i) and ii). The first two conditions i) and ii) are easily satisfied for a set $$A$$ consisting of the elements of a sequence in $$B\setminus \{0\}$$ converging to $$0$$ together with its limit point. However I do not know how to construct an example of a closed set $$A$$ where in addition iii) holds.

Note that for any integer $$n>0$$ there is a component $$C_n\subset A$$ that contains two points $$x_n$$ and $$y_n$$ such that $$|x_n|=\tfrac1n$$ and $$|y_n|=1$$. Recall that $$C_n$$ is a closed set.
Pass to a subsequence of $$C_n$$ that converges in the sense of Hausdorff; denote its limit by $$C_\infty$$. We may assume that $$y_n\to y_\infty$$.
Observe that $$C_\infty$$ is a closed connected subset of $$A$$ that contains $$0$$ and $$y_\infty$$ --- a contradiction.