Submetacompactness and weak submetacompactness in countable products

It will be shown that if Yi is a regular submetacompact DC-like space for each i ∈ ω, then Πi∈ω Yi is submetacompact. We also show that if Z is a hereditarily weakly submetacompact space and Yi is a regular weakly submetacompact DC-like (C-scattered) space for each i ∈ ω, then Z × Πi∈ω Yi weakly submetacompact.