# Limit of Intersection of Three Closed Sets

Let $$A_n$$, $$B_n$$ and $$C$$ be closed subsets in euclidean space such that $$\begin{equation} A_n \cap B_n \neq \varnothing, A_n \cap B_n \cap C\neq \varnothing \end{equation}$$ for each $$n=1,2,...$$, and $$\lim_{n\to \infty}A_n=A$$, $$\lim_{n\to \infty}B_n=B$$. If $$A\cap C\neq \varnothing$$, $$B\cap C\neq \varnothing$$ and $$A\cap B \neq \varnothing$$, then can we have $$A\cap B\cap C \neq \varnothing$$?

• What is your definition of $\lim A_n=A$? Do you want an example where $A \cap B \cap C$ is not empty or do you want to prove that this set cannot be empty? – Kavi Rama Murthy Nov 22 '19 at 5:40
• Your assumptions $A_n\cap B_n\ne\varnothing$ and $A_n\cap B_n\cap C\ne\varnothing$ are redundant, as the latter implies the former. – bof Nov 22 '19 at 6:45
• @Kabo Murphy Consider a correspondence $F(x)$, where $x\in E^n$ and $F(x) \in E^m$, the limit notation here is that when $x_n \to x$, $F(x_n) \to F(x)$, imagining $A_n$, $B_n$ as some $F_1(x_n)$ and $F_2(x_n)$. I want to prove this set is not empty... – Huaixin Nov 22 '19 at 12:57

If
$$A_n=\{-1\}\cup\{-2\}\cup[n,\infty)$$,
$$B_n=\{-1\}\cup\{-3\}\cup[n,\infty)$$,
$$C=\{-2\}\cup\{-3\}\cup[0,\infty)$$,
then:
$$A_n,B_n,C$$ are closed subsets of $$\mathbb R$$;
$$A_n\cap B_n=\{-1\}\cup[n,\infty)\ne\varnothing$$;
$$A_n\cap B_n\cap C=[n,\infty)\ne\varnothing$$;
$$A=\lim_{n\to\infty}A_n=\{-1,-2\}$$;
$$B=\lim_{n\to\infty}B_n=\{-1,-3\}$$;
$$A\cap C=\{-2\}\ne\varnothing$$;
$$B\cap C=\{-3\}\ne\varnothing$$;
$$A\cap B=\{-1\}\ne\varnothing$$;
$$A\cap B\cap C=\varnothing$$.