Topological–algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by C λ ( X ) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological–algebraic properties of C λ ( X ) . Our main results state that (1) C λ ( X ) is a topological vector space (a topological group) iff λ is a family of C-compact sets and C λ ( X ) = C λ ′ ( X ) , where λ ′ consists of all C-compact subsets of every set of λ. In particular, if C λ ( X ) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group C λ ( X ) is ω-narrow iff λ is a family of metrizable compact subsets of X.