We prove by using well-founded trees that a countable product of supercomplete spaces, scattered with respect to Čech-complete subsets, is supercomplete. This result extends results given in [Alstera], [Friedlera], [Frolika], [HohtiPelantb], [Pelanta] and its proof improves that given in [HohtiPelantb].
We solve the long standing problem of characterizing the class of strongly Fréchet spaces whose product with every strongly Fréchet space is also Fréchet.