κ-Fréchet Urysohn property of

For a Tychonoff space X, we denote by C k ( X ) the space of all real-valued continuous functions on X with the compact open topology. A space X is said to be κ-Fréchet Urysohn if for every open subset U of X and every x ∈ U ¯ , there exists a sequence { x n } n ∈ ω ⊂ U converging to x. In this paper, we show that C k ( X ) is κ-Fréchet Urysohn iff every moving off family of compact subsets of X has a countable subfamily which is strongly compact-finite. In particular, we obtain that every stratifiable Baire space C k ( X ) is an M 1 -space.