Comparing functional countability and exponential separability

Abstract: A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponentially separability. We start by studying these properties in GO spaces, where we extend results by Tkachuk and Wilson, and prove a conjecture by Dow. We also study some subspaces of products that are functionally countable and the influence of the $G_\delta$-topology on exponential separability. Finally, we give some examples of functionally countable spaces that are separable and uncountable.