Some upper bounds for density of function spaces

Abstract

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 which is closed under finite unions. We proved that the density of the space C α ( X , Y ) is at most iw ( X ) ⋅ d ( Y ) where iw ( X ) denotes the i-weight of the Tychonoff space X, and d ( Y ) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space C k ( [ 0 , 1 ] , Y ) is 2 κ .