On weakening tightness to weak tightness (Q2173227)

In [Topology Appl. 249, 103--111 (2018; Zbl 1403.54002)] the second author introduced the notion of weak tightness, as follows: \(wt(X)\le\kappa\) means that \(X\) has a cover \(\mathcal{C}\) of cardinality at most \(2^\kappa\), such that every member, \(C\), has tightness at most \(\kappa\) and satisfies \(X=\bigcup\{\operatorname{cl}A:A\subseteq C\) and \(|A|\le2^\kappa\}\). If \(2^{\aleph_0}=2^{\aleph_1}\) then the Cantor cube \(2^{\omega_1}\) provides an example of a compact space of uncountable tightness and countable weak tightness; the existence of a ZFC example is left as an open problem. A homogeneous compact space satifies \(w(X)\le2^{wt(X)}\), this should be compared with the inequality \(|X|\le2^{t(X)}\) from [\textit{R. de la Vega}, ibid. 153, No. 12, 2118--2123 (2006; Zbl 1098.54002)]. The paper provides other inequalities that mirror results proved earlier for tightness under weaker assumptions.