Some unsolved problems concerning countably compact spaces

As is well known, the concept of countable compactness played an important role in much early work in general topology and was only ousted from its central position after A. Tychonoff showed that the more restrictive property of compactness was productive. Nevertheless present estimates of the gap between countable compactness and compactness cannot be very precise, for in each of the following questions if "countably compact" is replaced by "compact", then it is known that the resulting question has an affirmative answer. Yet the questions, as stated, appear to us to be quite challenging.