Some cardinal invariants on the space Cα(X,Y)

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 spaceCα(X,Y)is countable.(iii)Cα(X,Y)is of pointwise countable type.(iv)There exists a compact subset K ofCα(X,Y)such that π-character of K in the spaceCα(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 ofCα(X,Y)such that each sequence{gn:n∈ω}withgn∈Onfor eachn∈ω, has a cluster point inCα(X,Y).