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 $\ddot{I}{„}^{\ast }$ of the natural topology of the real line $\mathbb{R}$ with properties such that the space $(\mathbb{R},\ddot{I}{„}^{\ast })$ has a hereditary nowhere dense tightness and it has no $\ddot{I}{‰}_1$-resolvable subspaces, whereas $\hat{I}”(\mathbb{R},\ddot{I}{„}^{\ast })=\mathfrak{c}$. 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 $L^{\ddot{I}}‰$ is c.c.c., then every submaximal dense subspace of $L^{\hat{I}}º$ has disjoint tightness. As a corollary, for every $\hat{I}º\ge \ddot{I}‰$ there is a Tychonoff submaximal space $X$ such that $|X|=\hat{I}”(X)=\hat{I}º$ and $X$ has disjoint tightness.

13 pages. Minor changes