In an article I am reading, they state a theorem from Victor Klee's "On Certain Intersection Properties of Convex Sets" and I am having trouble picturing this. The statement is:

Let $C$ and $C_1, ..., C_n$ be closed convex sets in a Euclidean space satisfying: (i) $C \cap \bigcap_{i \neq j}^n C_i \neq \emptyset$ for $j=1,...,n$, and (ii) $C \cap \bigcap_{i=1}^n C_i = \emptyset$. Then $C \nsubseteq \bigcup_{i=1}^n C_i$.

I want to say (i) $C$ and $C_i𝑖$ taken separately share at least some point. (ii) But, when we take the intersection of $C_i$, there are not points shared among any sets. Then, this means that $C$ is not a subset of the union of all $𝐶_𝑖$.

other$C_i$, that is, for $i\neq j$. $\endgroup$ – Joel David Hamkins Feb 15 '13 at 17:52