On resolvability and tightness in uncountable spaces

We investigate connections between resolvability and different forms of tightness. This study is adjacent to [1,2]. We construct a non-regular refinement ${\tau }^{\ast }$ of the natural topology of the real line $\mathbb{R}$ with properties such that the space $(\mathbb{R},{\tau }^{\ast })$ has a hereditary nowhere dense tightness and it has no ${\omega }_1$-resolvable subspaces, whereas $\mathrm{\Delta }(\mathbb{R},{\tau }^{\ast })=\mathfrak{c}$. We also show that the proof of the main result of [1], being slightly modified, leads to the following strengthening: if $L$ is a Hausdorff space of countable character and the space $L^{\omega }$ is c.c.c., then every submaximal dense subspace of $L^{\kappa }$ has disjoint tightness. As a corollary, for every $\kappa \ge \omega$ there is a Tychonoff submaximal space $X$ such that $\mathrm{\mid }X\mathrm{\mid }=\mathrm{\Delta }(X)=\kappa$ and $X$ has disjoint tightness.