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If / is a continuous surjection from a normal space X onto a regular space Y , then there are a space Z and a perfect map bf : Z —» Y extending / such that X C Z C sX . If / is a continuous surjection from normal X onto Tychonov Y and sX\X is sequential, then Y is normal. More generally, if / is a continuous surjection from normal X onto regular Y and sX\X has the property that countably compact subsets are closed (this property is called C-closed), then Y is normal. There is an example of a normal space X such that sX\X is C-closed but not sequential. If X is normal and sX\X is first countable, then sX\X is locally compact. We began the study of the class of spaces ACRIN (all continuous regular images normal) in (FL), where we showed that if X is a normal space such that sX\X is finite, then X is ACRIN. Here, we give more general conditions on sX\X which imply that a normal space is ACRIN. For a Tychonov space X, we denote the Stone-Cech compactification of X by sX. We call a space Z such that X c Z c sX an intermediate space. If X and Y are Tychonov spaces and / : X —> Y is a continuous map, we denote by sf : sX —* sY the Stone extension of /. Let us recall some well-known properties of maps. 1. Lemma. Let f : X —► Y be a continuous surjection. (a) If X is normal and f is a closed mapping, then Y is normal. (b) If X is countably compact, then Y is countably compact. (c) // / is perfect and Y is countably compact, then X is countably compact.

Stone-Čech remainders which make continuous images normal