On dimensional properties of topological products

For any n = 2 , 3 , … , there exist a metrizable compactum Φ n and a compactum Y n such that dim Φ n ( = ind Φ n ) = dim Y n = ind Y n = n and dim ( Φ n × Y n ) = n + 1 < 2 n = ind ( Φ n × Y n ) . For any Tychonoff space X and any metrizable space Y, we have: d X × Y ⩽ ind X + dim Y and d X × Y ⩽ ind X + ind Y if, additionally, Y is strongly metrizable (in particular, strongly paracompact, or separable, or compact).