The Stacks project

110.2 An empty limit

This example is due to Waterhouse, see [Waterhouse]. Let $S$ be an uncountable set. For every finite subset $T \subset S$ consider the set $M_ T$ of injective maps $T \to \mathbf{N}$. For $T \subset T' \subset S$ finite the restriction $M_{T'} \to M_ T$ is surjective. Thus we have an inverse system over the directed partially ordered set of finite subsets of $S$ with surjective transition maps. But $\mathop{\mathrm{lim}}\nolimits M_ T = \emptyset $ as an element in the limit would define an injective map $S \to \mathbf{N}$.