Example 1

If the $E_\alpha$ are finite sets, Theorem 1 can be applied by taking $\mathfrak{S}_\alpha$ to be the set of all subsets of $E_\alpha$. \* This example is generalized in General Topology to the situation in which the $E_\alpha$ are compact topological spaces, the $f_{\alpha\beta}$ continuous maps, and $\mathfrak{S}_\alpha$ the set of closed subsets of $E_\alpha$ (General Topology, Chapter I, § 9, no. 6). \*