# $\bigcap_{i \in I}\overline{co}(A_i)=\overline{co}\left(\bigcap_{i \in I}\overline{A_i}\right)$

Let $$(A_i: i \in I)$$ be a family of sets in a topological vector space such that for all $$i,j \in I$$ there exists $$k \in I$$ for which $$A_i \cap A_j=A_k$$. Denote by $$\overline{co}(X)$$ the closure of the convex hull of $$X$$. Is it true that $$\bigcap_{i \in I}\overline{co}(A_i)=\overline{co}\left(\bigcap_{i \in I}\overline{A_i}\right)\,\,\,?$$

One inclusion: Since $$\overline{A_i} \subseteq \overline{co}(A_i)$$ for all $$i$$ then $$\bigcap_i \overline{A_i} \subseteq \bigcap_i \overline{co}(A_i)$$. Therefore $$\overline{co}\left(\bigcap_i \overline{A_i}\right) \subseteq \overline{co}\left(\bigcap_i \overline{co}(A_i)\right)=\bigcap_i \overline{co}(A_i).$$

Comment 1. The hypothesis on the family $$\{A_i:i \in I\}$$ is necessary, see Kavi's answer.

Comment 2. The claim is easily seen to be verified if $$\{A_i: i \in I\}$$ has a minimum, which holds, e.g., if $$I$$ is finite.

Comment 3. The claim $$\bigcap_{i}\overline{co}(A_i)=\overline{co}\left(\bigcap_{i}A_i\right)$$ (i.e., replacing $$\overline{A_i}$$ with $$A_i$$ in the right side) is false. As a counterexample, let $$A_i=(0,2^{-i})\cup\{1\}$$ for all integers $$i\ge 1$$. Then the left side is $$[0,1]$$ and the right side is $$\{1\}$$.

The claim is false. For each positive integer $$n$$ let $$A_n$$ be the set of integers with absolute value greater than $$n$$. Then the left hand side is $$\mathbf{R}$$ and the right hand side is $$\emptyset$$.
Let $$A=\{1,4\}, B=\{2,3\}$$. The RHS is empty. But $$co(A)=[1,4],co(B)=[2,3]$$ so $$2 \in LHS$$.