On the finite-dimensionality of topological products

It is proved that there exist integers e(k, l) ⩾ − 1 for k, l = − 1, 0, 1, … such that Ind X × Y ⩽ e(Ind X, Ind Y) if the space X × Y is normal (and Hausdorff), Y is locally compact paracompact (in particular, compact) and Ind X < ∞, Ind Y < ∞ (therefore any normal product of two finitedimensional in the sense of Ind spaces, one of which is locally compact paracompact is finite-dimensional in the same sense). Analogous assertions hold for any strongly paracompact product, any normal product with one metrizable factor and any normal product of a pseudocompact space and a k-space. Also it is proved that a strongly paracompact or a z-embedded subspace of a finitedimensional in the sense of Ind normal space is finite-dimensional in the same sense.