ABSTRACT
All spaces considered are Hausdorff. It is well known that the existence of first countable, countably compact, non-compact perfectly normal spaces is independent with the usual axioms of set theory. W. Weiss [9] showed that Martin’s Axiom plus the negation of CH implies that each first countable, countably compact perfectly normal space is compact. Ostaszewski [5] using constructs a non-compact example. It is still unknown whether there is in ZFC a separable, first countable, countably compact non-compact space. These have been constructed using special axioms however. Vaughn [8] has constructed a first countable non-normal countably compact space in ZFC which is also ω-bounded, hence obviously not separable. It seems natural to strengthen the compactness requirement in these 104questions. For a cardinal κ, a space is initially κ-compact if each open cover with at most κ elements has a finite subcover. Is there a non-compact first countable initially w1-compact space? The answer to this question is unknown but we show that the answer is no if we assume CH. We shall use the notation and terminology of the Engelking text [4]. As usual cardinals are identified with initial ordinals.
