We introduce and investigate statistical convergence in topological and uniform spaces and show how this convergence can be applied to selection principles theory, function spaces and hyperspaces.

We study statistical versions of several classical kinds of convergence of sequences of functions between metric spaces (Dini, Arzelà, and Alexandroff) in different function spaces. Also, we discuss a statistical approach to recently introduced notions of strong uniform convergence and exhaustiveness.

Abstract. We continue the investigation of star selection principles first considered in [9]. We are concentrated onto star versions of the Hurewicz covering property and star selection principles related to the classes of open covers which have been recently introduced.2000 AMS Classification: 54D20.

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Abstract. We use Ramseyan partition relations to characterize:• the classical covering property of Hurewicz;• the covering property of Gerlits and Nagy;• the combinatorial cardinal numbers b and add(M).Let X be a T 3 1 2 -space. In [9] we showed that C p (X) has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the GerlitsNagy covering property. Now we show that the following are equivalent:1. C p (X) has countable fan tightness and the Reznichenko property. 2. All finite powers of X have the Hurewicz property.We show that for C p (X) the combination of countable fan tightness with the Reznichenko property is characterized by a Ramseyan partition relation. Extending the work in [9], we give an analogous Ramseyan characterization for the combination of countable strong fan tightness with the Reznichenko property on C p (X). Introduction.We continue the investigation of combinatorial properties of open covers. The main theme in this paper, as in the previous ones, will be to show that certain concepts appearing in mathematical literature can be characterized by simple selection principles. These selection principles are then characterized game-theoretically and Ramsey-theoretically.To ease the reader into our topic we recall the important basic definitions here in the introduction. Directly after that we give in Section 1 an abstract setting for the typical arguments used to derive Ramsey-theoretic results from game-theoretic circumstances. This, once and for all, will codify arguments that have repeatedly shown up in the study of selection princi-