Some cardinal invariants on the space

Let C α ( 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 ( E 1 ) and ( E 2 ) on the triple ( α , X , Y ) which yield new equalities and inequalities between some cardinal invariants on C α ( X , Y ) and some cardinal invariants on the spaces X, Y such as: Theorem s an equiconnected space with a base consisting of φ-convex sets, then for each f ∈ C ( X , Y ) , χ ( f , C α ( X , Y ) ) = α a ( X ) . w e ( f ( X ) ) . ary be a noncompact metric space and let the triple ( α , X , Y ) satisfy ( E 1 ) . The following are equivalent: (i) X , Y ) is a first-countable space. acter of the space C α ( X , Y ) is countable. X , Y ) is of pointwise countable type. exists a compact subset K of C α ( X , Y ) such that π-character of K in the space C α ( X , Y ) is countable. X ) ⩽ ℵ 0 . X , Y ) is metrizable. X , Y ) is a q-space. exists a sequence { O n : n ∈ ω } of nonempty open subset of C α ( X , Y ) such that each sequence { g n : n ∈ ω } with g n ∈ O n for each n ∈ ω , has a cluster point in C α ( X , Y ) .