AbstractLet Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X such that closed under finite unions. We define two properties (E1) and (E2) on the triple (α,X,Y) which yield new equalities and inequalities between some cardinal invariants on Cα(X,Y) and some cardinal invariants on the spaces X, Y such as: TheoremIf Y is an equiconnected space with a base consisting of φ-convex sets, then for each f∈C(X,Y), χ(f,Cα(X,Y))=αa(X).we(f(X)).CorollaryLet Y be a noncompact metric space and let the triple (α,X,Y) satisfy (E1). The following are equivalent: (i)Cα(X,Y) is a first-countable space.(ii)π-character of the space Cα(X,Y) is countable.(iii)Cα(X,Y) is of pointwise countable type.(iv)There exists a compact subset K of Cα(X,Y) such that π-character of K in the space Cα(X,Y) is countable.(v)αa(X)⩽ℵ0.(vi)Cα(X,Y) is metrizable.(vii)Cα(X,Y) is a q-space.(viii)There exists a sequence {On:n∈ω} of nonempty open subset of Cα(X,Y) such that each sequence {gn:n∈ω} with gn∈On for each n∈ω, has a cluster point in Cα(X,Y).