On the density and products of spaces (Q1985630)

Extending an earlier result of the author to higher cardinalities it is shown that for a family \(\{X_\alpha \mid \alpha\in 2^\kappa\}\) of spaces each of density \(\kappa\), a cardinal \(>\omega\), the product space \(\prod_{\alpha\in2^\kappa}X_\alpha\) contains a dense subset \(D\) of cardinality \(\kappa\) such that for every \(C\subset D\) of cardinality \(\kappa\) there is \(\alpha\in 2^\kappa\) such that the projection of \(C\) onto \(X_\alpha\) is dense in \(X_\alpha\).