4 · See which coordinates a map can read

A product may have a nonzero integer in every coordinate. Specker's theorem says that an additive map from a countable product to the integers can read only finitely many of them.

Try this. Choose how many coordinates the map reads. Change one coordinate inside that range and one outside it. Only a coordinate inside the range can change the output.

The finite picture illustrates Specker's conclusion but cannot prove the infinite theorem. Together with Nöbeling's theorem, it gives the product description of the free solid group in Corollary 5.5.