Baire 1 Functions and the Topology of Uniform Convergence on Compacta

Let X be a Tychonoff topological space, B 1 ( X , R ) be the space of real-valued Baire 1 functions on X and τ U C be the topology of uniform convergence on compacta. The main purpose of this paper is to study cardinal invariants of ( B 1 ( X , R ) , τ U C ) . We prove that the following conditions are equivalent: (1) ( B 1 ( X , R ) , τ U C ) is metrizable; (2) ( B 1 ( X , R ) , τ U C ) is completely metrizable; (3) ( B 1 ( X , R ) , τ U C ) is Čech-complete; and (4) X is hemicompact. It is also proven that if X is a separable metric space with a non isolated point, then the topology of uniform convergence on compacta on B 1 ( X , R ) is seen to behave like a metric topology in the sense that the weight, netweight, density, Lindelof number and cellularity are all equal for this topology and they are equal to c = | B 1 ( X , R ) | . We find further conditions on X under which these cardinal invariants coincide on B 1 ( X , R ) .