On resolvability, connectedness and pseudocompactness

Abstract

We prove that: I. If $L$ is a $T_1$ space, $|L|>1$ and $d(L)\le \kappa \ge \omega$, then there is a submaximal dense subspace $X$ of $L^{2^{\kappa }}$ such that $|X|=\Delta (X)=\kappa$; II. If $\mathfrak{c}\le \kappa ={\kappa }^{\omega }<\lambda$ and $2^{\kappa }=2^{\lambda }$, then there is a Tychonoff pseudocompact globally and locally connected space $X$ such that $|X|=\Delta (X)=\lambda$ and $X$ is not ${\kappa }^{+}$-resolvable; III. If ${\omega }_1\le \kappa <\lambda$ and $2^{\kappa }=2^{\lambda }$, then there is a regular space $X$ such that $|X|=\Delta (X)=\lambda$, all continuous real-valued functions on $X$ are constant (so $X$ is pseudocompact and connected) and $X$ is not ${\kappa }^{+}$-resolvable.

arXiv comment12 pages, no figures, minor changes