  Using a dictionary translating a variety of classical and modern covering
properties into combinatorial properties of continuous images, we get a simple
way to understand the interrelations between these properties in ZFC and in the
realm of the trichotomy axiom for upward closed families of sets of natural
numbers. While it is now known that the answer to the Hurewicz 1927 problem is
positive, it is shown here that semifilter trichotomy implies a negative answer
to a slightly weaker form of this problem.
