On strong measure zero subsets of \({}^\kappa 2\) (Q2773247)

The Cantor set (\(2^\omega\)) and the irrationals (\(\omega^\omega\)) when viewed as spaces of functions with domain \(\omega\) have natural \(\kappa\)-generalizations to uncountable regular cardinals. Specifically give \({}^\kappa 2\) and \({}^\kappa\kappa\) the topologies in which the basic open sets have the form \([\eta]=\{ f: \eta\subset f\}\) for each function \(\eta\) with \(\text{Dom}(\eta)\in \kappa\). Therefore such properties as \(\kappa\)-meager and \(\kappa\)-Baire are natural to study. In addition, a subset \(A\) of \({}^\kappa 2\) or \({}^\kappa \kappa\) is said to be of strong measure zero, if for each cofinal \(X\subset \kappa\), there is a sequence \(\langle \eta_\xi : \xi \in X\rangle\) such that \(\eta_\xi\in {}^\xi\kappa\) and \(A\subset \bigcup \{[\eta_\xi ] : \xi \in X\}\). The paper first develops nice equivalences for the property of being of strong measure zero. It then relates this notion to the \(\kappa\) analogues of the bounding number, \(\mathfrak b_\kappa\), and the dominating number \(\mathfrak d_\kappa\). It is shown to be undecidable as to whether the ideal of strong measure zero sets is \(2^\kappa\)-additive. In the third section, there is a nice construction (3.5) which can be used to establish, (3.10), that the Generalized Borel Conjecture, \(\text{GBC}(\kappa)\), fails of successor \(\kappa\) which satisfy \(\kappa^{<\kappa}=\kappa\). The statement \(\text{GBC}(\kappa)\) of course says that every strong measure zero set has cardinality at most \(\kappa\).