On resolvability, connectedness and pseudocompactness

We prove that:I. If L is a \(T_1\) $T_1$ space, \(|L|>1\) $|L|>1$ and \(d(L) \leq \kappa \geq \omega\) $d(L)\le \kappa \ge \omega$ , thenthere is a submaximal dense subspace X of \(L^{2^\kappa}\) $L^{2^{\kappa }}$ such that \(|X|=\Delta(X)=\kappa\) $|X|=\mathrm{\Delta }(X)=\kappa$ . II. If \(\mathfrak{c}\leq\kappa=\kappa^\omega<\lambda\) $\mathfrak{c}\le \kappa ={\kappa }^{\omega }<\lambda$ and \(2^\kappa=2^\lambda\) $2^{\kappa }=2^{\lambda }$ , then there is a Tychonoff pseudocompact globally and locally connected space X such that \(|X|=\Delta(X)=\lambda\) $|X|=\mathrm{\Delta }(X)=\lambda$ and X is not \(\kappa^+\) ${\kappa }^{+}$ -resolvable. III. If \(\omega_1\leq\kappa<\lambda\) ${\omega }_1\le \kappa <\lambda$ and \(2^\kappa=2^\lambda\) $2^{\kappa }=2^{\lambda }$ , then there is a regular space X such that \(|X|=\Delta(X)=\lambda\) $|X|=\mathrm{\Delta }(X)=\lambda$ , all continuous real-valued functions on X are constant (so X is connected) and X is not \(\kappa^+\) ${\kappa }^{+}$ -resolvable.