On resolvability and tightness in uncountable spaces

Abstract

We investigate connections between resolvability and different forms of tightness. This study is adjacent to [1,2]. We construct a non-regular refinement $τ^*$ of the natural topology of the real line $\mathbb{R}$ with properties such that the space $(\mathbb{R}, τ^*)$ has a hereditary nowhere dense tightness and it has no $ω_1$-resolvable subspaces, whereas $Δ(\mathbb{R}, τ^*) = \frak{c}$. 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 $L^ω$ is c.c.c., then every submaximal dense subspace of $L^κ$ has disjoint tightness. As a corollary, for every $κ\geq ω$ there is a Tychonoff submaximal space $X$ such that $|X|=Δ(X)=κ$ and $X$ has disjoint tightness.