On certain groups of functions

Let C ( X , G ) denote the group of continuous functions from a topological space X into a topological group G with the pointwise multiplication and the compact-open topology. We show that there is a natural topology on the collection of normal subgroups Δ ( X ) of C ( X , G ) of the M p = { f ∈ C ( X , G ) : f ( p ) = e } which is analogous to the hull-kernel topology on the commutative Banach algegra C ( X ) of all continuous real or complex-valued functions on X . We also investigate homomorphisms between groups C ( X , G ) and C ( Y , G ) .