The o-Malykhin property for spaces

Let X be a Tychonoff space and C k ( X ) be the vector topological space of continuous real-valued functions on X with the compact open topology. The problem of characterizing C k ( X ) in terms of X has interested several authors; in particular, Gruenhage and Ma for the baireness property and recently Sakai with the κ-Fréchet Urysohn property. Motivated by their works, we are interested in the o-Malykhin property for C k ( X ) . In this note, we will show that C k ( X ) is o-Malykhin if and only if every moving off collection of non-empty compacts of X contains an infinite compact-finite collection. We will also characterize the o-Malykhin property for C k ( X ) by a topological game defined on X, and we will give some related results.