Countable paracompactness of σ-products

We present an example of a σ-product that is not countably paracompact but all of whose finite subproducts are countably paracompact. This example also shows that countable paracompactness of a σ-product may depend on the choice of base point. We also show that normal non-trivial σ-products are countably paracompact, improving a result of Chiba. Finally we give a new proof that σ-products of ordinals at base point 0 are κ-normal and strongly zero-dimensional.