Given Hausdorff spaces X and Y and a perfect irreducible and θ-continuous map f from X onto Y, technique that carries open (ultra) filters on X to open (ultra) filters on Y back and forth in a natural way is introduced. It is proved that if f is a perfect irreducible and θ-continuous map from X onto Y, then X is almost realcompact if and only if Y is almost realcompact. Several commutativity relations between the ‘absolutes of almost realcompactifications’ and the ‘almost realcompactifications of absólutes’ of a space X are discussed.
