Tri-quotient maps become inductively perfect with the aid of consonance and continuous selections

Generalizing the result of Arhangelʹskii that each open map with Čech-complete domain is compact-covering, it is proved that every tri-quotient map with consonant domain is harmonious, thus compact-covering, and its range is consonant. The latter constitutes a strong answer to a question of Nogura and Shakhmatov. Conditions for harmonious maps to be inductively perfect, or countable compact-covering and for countable compact-covering maps to be harmonious are given. They extend theorems of Just and Wicke.