Sequential order under MA (Q1763615)

Set the cardinal number \(\mathfrak{b}=\min \left\{ \left| B\right| :B \text{ is an unbounded set of functions from }\omega \text{ into }\omega \right\} \). As usual, ``unbounded'' is with respect to the order ``less than except for finitely many points.'' It is well known that if Martin's Axiom is true, then \(\mathfrak{b}=\mathfrak{c}\). In this paper, the author uses the assumption that \(\mathfrak{b}=\mathfrak{c}\) to construct a compact sequential space with sequential order \(4\). The space is constructed as the Stone space of a certain Boolean algebra, but it ``looks like'' an iteration of a \(\psi \)-like construction. It was shown earlier by Bashkirov that if the Continuum Hypothesis assumed, then one can construct compact sequential spaces of sequential orders up to \(\omega _{1}\), the upper limit for sequential spaces. This may be a step toward removing the \( CH\) assumption from Bashkirov's result, or perhaps it is a step toward determining whether under some assumption like \(PFA\), there is a bound on the sequential order of compact spaces of countable tightness.