Spaces constructed using ♣ and weaker related axioms
Keywords:
compact, countably compact, $\omega_1$-compact, $\sigma$-countably compact, club set, stationary setAbstract
Some very simple locally compact, first countable, Hausdorff spaces are constructed using \% and some major weakenings. The first of these examples is relevant to a wide-open problem in set-theoretic topology:
Problem. What is the least cardinality of a locally compact, first countable Hausdorff (hence Tychonoff) space that is $\omega_1$-compact, yet not $\sigma$-countably compact?
The first example is a witness to this cardinality being consistently $\aleph_1$.
The other two main examples are $2-1$ closed preimages of $\omega_1$ constructed in a similar way. They illustrate how suitable \% is to show, in a simple way, that the following powerful axiom is not a consequence of ZFC.
Axiom 1. Every first countable, countably compact Hausdorff space is either compact or contains a copy of $\omega_1$.
A simple criterion is shown for 2-1 closed continuous preimages of $\omega_1$ to not contain a copy of $\omega_1$ : every uncountable subset must have the fibers (point-inverses) over a stationary set in its closure. Major weakenings of \%, including club guessing, are shown to be adequate to produce such spaces. A further weakening, $\mho_2$, is shown to be equivalent to a topologically defined subclass of such spaces without copies of $\omega_1$. These weaker axioms are independent of CH , but also compatible with major strengtheings of MA $+\neg \mathrm{CH}$, and help to gauge the strength of Axiom 1, which follows from the Proper Forcing Axiom (PFA) and is also compatible with CH .
Various axioms are discussed that negate the existence of such simple spaces as the three main examples, with respect to them either being counterexamples to Axiom 1, or being of cardinality $\aleph_1$ and locally compact, $\omega_1$-compact, and not $\sigma$-countably compact.
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