On two consequences of CH established by Sierpiński

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on uncountable subsets of the real line. The subject is closely related  to the existence and properties of Lusin sets and  K-Lusin sets in the Baire space N^N, i.e., uncountable sets in N^N intersecting each compact set in N^N in an at most countable set.

The results come from two joint papers with Roman Pol, available at
http://arxiv.org/abs/2306.11712     and     http://arxiv.org/abs/2403.03110
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