Let $D = [c,d]\times [c,d]\subseteq \mathbb{R}^2$ and let $A$ be the set of all closed subsets of $D$. For $a \in D$ and $B\in A,$ define $d(a,B) := \min\{d(a,b) | b\in B\},$ where the $d$ inside the min is the Euclidean metric on $\mathbb{R}^2$. For $B,C \in A,$ let $d_A(B,C) = \max\{\max_{b\in B} d(b,C), \max_{c\in C} d(c,B)\}.$ For a metric space $X$, define an isolated point $x\in X$ to be a point such that $\exists r > 0$ so that $B^*(x, r) \cap X = \emptyset,$ where $B^*(x,r) = B(x,r)\backslash \{x\}$ and $B(x,r)$ is the open ball centered at $x$ of radius $r$.

Prove that the set $G := \{B\in A: B\text{ has no isolated points}\}$ is residual in $(A,d_A)$ by showing that for $k\geq 1,$ the set $F_k := \{B \in A : \exists b \in B, \overline{B}(b, \frac{1}k)\cap B = \{b\}\}$ is closed and nowhere dense. Here, $\overline{B}(b, \frac{1}k)$ is the closed ball of radius $\frac{1}k$ centered at $b$.

If one shows that the sets $F_k$ are closed and nowhere dense, then since $G^C = \cup_{n\geq 1} F_n$ (which is not hard to show), $G^C$ is first category.

However, I'm really struggling to show that the sets $F_k$ are nowhere dense. I tried taking an element $C \in F_k^C$ and finding an $r > 0$ so that $B(C, r) \subseteq F_k^C, B(C,r) := \{B \in A : d_A(B,C) < r\}.$ I think $r$ should depend on $\sup_{a,b \in C} d(a,b)$ and on $\frac{1}k,$ but I don't know how to define $r$ to make this work. Assume a working r is chosen (I know it exists, but I'm not sure what it is). Let $D \in B(C,r).$ I tried seeing if I could choose r (independent of $D$ of course but possibly dependent on $C$) to get a contradiction if I assumed that $D \in F_k.$ But I wasn't able to do so.

In particular, if I assume $D \in F_k$ and choose $b \in D$ so that $\overline{B}(b, \frac{1}n)\cap D = \{b\},$ my issue is that $b$ might very well be in $C$.

Alternatively, I tried showing that every sequence with elements in $F_k$ that converges in $A$ also converges in $F_k,$ but I wasn't able to show this either.

If I can show the sets are closed, then to show they're nowhere dense it suffices to show that they have empty interiors. But given an element $C \in F_k,$ I'm not sure how to find for every $r > 0$ a set $B$ so that $d_A(B,C) < r$ but $B\not\in F_k.$

I know some things that might be useful: the EVT on compact metric spaces, the fact that for a compact metric space, for all sets $K$ of closed sets in $X$, if every finite subset of $K$ has nonempty intersection, so does $K$. Also, every compact metric space is complete and totally bounded.

As a side note, I know I've posted this question before with the exact same background information. However, the requirements for this question are quite different from the requirements in the previous question I posted so I think an answer to this question will be quite a bit different to an answer to the previous question.


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