# Is there in ZFC a topological space which is normal, ccc, countably compact, first countable and non-compact?

I am looking for a space as in the title and since many very similar spaces do exist in the literature, I wonder whether someone has a reference (different from the ones I cite below) or just some remarks about this question.

A space is ccc iff any family of pairwise disjoint open sets is at most countable, and countably compact iff any countable open cover has a finite subcover. A space is separable iff it has a countable dense subset. It is immediate that a separable space is ccc.

Related facts that I know:

• If one looks for a perfectly normal such space, the answer is independent of ZFC: MA + $\neg$CH implies that a perfectly normal countably compact space is compact (Weiss 1978), and this result is also compatible with CH (Eisworth 2002). Moreover, if $\diamondsuit$ holds, there is a perfectly normal, (hereditarily) separable, first countable, countably compact, non-compact space (Ostaszewski 1976).

• If one relaxes first countability, such spaces do exist in ZFC, see for instance Henno Brandsma's answer to this related question: http://mathoverflow.net/questions/79021 (the given examples are Frechet-Uryson and collectionwise normal). One can also obtain separable examples (Franklin-Rajadopalan 1970).

• If one wants a space which is only Tychonov but not normal, there is an example in ZFC (Bell 1989).

• The related question of whether there is a countably compact, first countable, separable, non-compact regular space in ZFC is still open, as far as I know. Nyikos wrote several papers on the subject.

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I'm guessing you already checked Counterexamples in Topology? –  David White Mar 13 '13 at 22:22
Actually, I just browsed it without a careful checking, since Bell's example was published 10 years after Counterexamples in Topology. –  Mathieu Baillif Mar 13 '13 at 22:59
It's been a while since I've thought about these things, but I do have a couple of comments: In the work of mine you mention above, normality is a bit of a red herring as it is "perfect" and "regular" that allow us to build the notion of forcing needed for the proof to go through. It is also consistent with ZFC that every first countable, countably compact regular space is either compact or contains a homeomorphic copy of $\omega_1$. (I think that's originally due to Fremlin and Nyikos, but Peter Nyikos and I had a paper in the 2000s showing this consistent with CH as well.) –  Todd Eisworth Mar 14 '13 at 2:55
Thanks for your comment, Todd. I thought that the result about first countable countably compact spaces containing a closed copy of omega_1 was originally proved by Balogh under the PFA, but I might be wrong. Actually, Franklin−Rajagopalan′s example can be built as the union of omega_1 and countable dense subset of isolated points. Also related is the result in ZFC that a regular, perfect and feebly compact space is first countable and ccc (I think it is due to Porter and Woods). And a feebly compact normal space is countably compact. –  Mathieu Baillif Mar 14 '13 at 13:10
Maybe it was Balogh. I know Fremlin and Nyikos were working on the problem as well, but that's all well before my time! –  Todd Eisworth Mar 14 '13 at 14:51