# Empty projective limit of finite sets

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.

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.