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Let $I$ be a preordered set and $(A_i)_{i\in I}$ be a collection of non-empty finite sets. I want to find a pair $(I,(A_i)_{i\in I})$ such that the projective limit $\varprojlim A_i$ is empty.

I know that if $I$ is also a directed set, then this is not possible. So, the first thing I tried was to find a preordered set which is not directed. I've found that $\mathbb{N}\coprod\mathbb{N}$, with $i\leq j$ if and only if $i,j$ live in the same copy of $\mathbb{N}$ and $i\leq j$ there, is an example of this. Nevertheless, I didn't found some sets $A_i$ with empty projective limit.

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Consider the following diagram: $A = \{0\}$, $B = \{1\}$, $C = \{0,1\}$ with connecting maps the inclusions $A\to C$ and $B\to C$. The limit of this diagram is the intersection of $A$ and $B$ in $C$, which is empty.

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