On resolvability, connectedness and pseudocompactness

Abstract

We prove that: I. If L is a \(T_1\) space, \(|L|>1\) and \(d(L) \leq \kappa \geq \omega\), then there is a submaximal dense subspace X of \(L^{2^\kappa}\) such that \(|X|=\Delta(X)=\kappa\). II. If \(\mathfrak{c}\leq\kappa=\kappa^\omega<\lambda\) 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 \(\omega_1\leq\kappa<\lambda\) 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 connected) and X is not \(\kappa^+\)-resolvable.