On weakening tightness to weak tightness

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property wt(X) ≤ t(X). We investigate several wellknown results concerning t(X) and consider whether they extend to theweak tightness setting. First we give an example of a non-sequential compactum X such that wt(X) = ℵ0 < t(X) under 2ℵ0 = 2ℵ1 . In particular, this demonstrates the celebrated Balogh’s (Proc Am Math Soc 105(3):755–764, 1989) Theorem does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then |X| ≤ 2wt(X)πχ(X).