On weakening tightness to weak tightness

Abstract

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property $wt(X)\le t(X)$. We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $wt(X)=\aleph _0<t(X)$ under $2^{\aleph _0}=2^{\aleph _1}$. 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 $|X|\le 2^{wt(X)\pi \chi (X)}$. This refines a theorem of de la Vega (Topol Appl 153:2118–2123, 2006). In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhász and van Mill (Proc Am Math Soc 146(1):429–437, 2018). In this connection we also show $w(X)\le 2^{wt(X)}$ for a homogeneous compactum. Third, we show that if X is a $T_1$ space, $wt(X)\le \kappa$, X is $\kappa ^+$-compact, and $\psi (\overline{D},X)\le 2^\kappa$ for any $D\subseteq X$ satisfying $|D|\le 2^\kappa$, then (a) $d(X)\le 2^\kappa$ and (b) X has at most $2^\kappa$-many $G_\kappa$-points. This is a variation of another theorem of Balogh (Topol Proc 27:9–14, 2003). Finally, we show that if X is a regular space, $\kappa =L(X)wt(X)$, and $\lambda$ is a caliber of X satisfying $\kappa <\lambda \le \left( 2^{\kappa }\right) ^+$, then $d(X)\le 2^{\kappa }$. This extends of theorem of Arhangel$'$skiĭ (Topol Appl 104:13–26, 2000).