Topological games in Ramsey spaces

Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property. In this article, we present a general overview of the combinatorial structure of topological Ramsey spaces and their main properties, and we propose an alternative proof of the abstract Ellentuck theorem for a large family of axiomatized topological Ramsey spaces. Additionally, we introduce the notion of selective axiomatized topological Ramsey space, and generalize Kastanas games in order to characterize the Ramsey property for this broad family of topological Ramsey spaces through topological games.