A study on special subsets of real numbers derived from local properties of function spaces and their critical cardinalities

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For a topological space X, we denote by Cp(X) the space of all continuous functions with the topology of pointwise convergence. It is known that a local property of Cp(X) can be characterized in terms of a covering property of X. For example, Gerlits and Nagy showed that Cp(X) is Frechet if and only if X is a γ-set. A γ-set is strong measure zero, so some local properties of Cp(X) are related with some singular sets of reals appeared in descriptive set-theory. We studied some relations between local properties of Cp(X) and singular sets X of reals, and answered some open problems posed by Scheepers, Bukovsky and so on.