A study on Scheepers' conjecture of special subsets of reals in view of a sequence of upper semi-continuous functions

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Concerning Scheepers’ conjecture on special subsets of the real line and pseudo-normal convergence of continuous functions in a function space Cp(X) with the topology of pointwise convergence, we tried to answer the conjecture in view of a sequence of upper semi-continuous functions. We found that a Pixley-Roy hyperspace is useful to consider Scheepers’ conjecture. We obtained a characterization for a Pixley-Roy hyperspace is Frechet-Urysohn or has the discrete countable chain condition. Moreover, concerning selective separability which is connected with Scheepers’ conjecture, we gave answers to some open problems.