# Showing a set is residual

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.