# Is compactness of sequentially compact sequential Hausdorff spaces determined by the well ordered space $\omega_1$?

The space $X$ is sequential if for each nonclosed $A \subset X$, there exists a convergent sequence $a_n \rightarrow x$ so that $a_n \in A$ but $x \notin A$. By $\omega_1$ we mean an uncountable well ordered set (with the order topology) so that each bounded subset is countable. Sequentially compact means each sequence in $X$ has a convergent subsequence.

Suppose $X$ is sequential, sequentially compact, and Hausdorff (or weakly Hausdorff). If $X$ is not compact, must $X$ contain a subspace homeomorphic to $\omega_1$?

(The converse is true. If $X$ is compact then $\omega_1$ cannot embed in $X$.)

Edit:

There is a well studied space of Ostaszewski which is plausibly? obviously? obviously not? a counterexample:

A. J. Ostaszewski, “On countably compact, perfectly normal spaces,” Journal of the London Mathematical Society, volume 14, (1976), pp. 505–516

It is apparent that attempting to employ sequential compactness in an effort to characterize the compactness property of sequential spaces can take us outside ZFC.

[Vaughan(1984)] J. E. Vaughan, “Countably compact and sequ entially compact spaces,” inHand-book of Set-Theoretic Topology (edited by K. Kunen and J. E. Vaughan), pp. 569–602

Corollary 6.12 of Anthony Goreham's arxived paper http://arxiv.org/pdf/math/0412558v1.pdf, asserts: The statement ‘Compactness and sequential compactness are equivalent for perfectly regular sequential spaces’ is independent of ZFC.

No. As suggested in the comments, in the example of Ostaszewski, X is sequential and sequentially compact, and each open set is countable or co-countable. No such space can contain $w_{1}$ with the order topology, since $w_{1}$ contains an uncountable closed subspace with uncountable complement.

On the other hand, every 1st first countable space is Frechet-Urysohn and in particular sequential. As noted in the following paper's abstract, there exists a model of ZFC + CH such that every 1st countable sequentially compact space is either compact or contains a copy of $w_{1}$ with the order topology.

T. Eisworth and P. Nyikos, First countable, countably compact spaces and the Continuum Hypothesis, Trans. Amer. Math. Soc. 357 (2005), 4329-4347.