Pseudoradial spaces: Separable subsets, products and maps onto Tychonoff cubes

We work around our question of whether compact non-pseudoradial spaces have separable such subspaces. We obtain results about products of pseudoradial spaces and obtain more conditions which guarantee that each compact sequentially compact space is pseudoradial. We also discuss some questions of Šapirovskiı̆ which are also directed at separating the non-separable from the separable. We reinforce the need to focus on the space Iω2.