Topological group criterion for C(X) in compact-open-like topologies, I

We address questions of when (C(X),+) is a topological group in some topologies which are meets of systems of compact-open topologies from certain dense subsets of X. These topologies have arisen from the theory of epimorphisms in lattice-ordered groups (in this context called “epi-topology”). A basic necessary and sufficient condition is developed, which at least yields enough insight to provide the general answer “sometimes Yes and sometimes No”. After some reduction the situation seems to become Set Theory (which view will be reinforced by a sequel to this paper “Topological group criterion for C(X) in compact-open-like topologies, II”).