Ohio completeness and products

In [A.V. Arhangel'skiĭ, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79–90], Arhangel'skiĭ introduced the notion of Ohio completeness and proved it to be a useful concept in his study of remainders of compactifications and generalized metrizability properties. We will investigate the behavior of Ohio completeness with respect to closed subspaces and products. We will prove among other things that if an uncountable product is Ohio complete, then all but countably many factors are compact. As a consequence, Rκ is not Ohio complete, for every uncountable cardinal number κ.