On resolvability, connectedness and pseudocompactness

Anton Lipin

Published 2023-08-02Version 1

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 $\frak{c}\leq\kappa=\kappa^\omega<\lambda$ 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 pseudocompact and connected) and $X$ is not $\kappa^+$-resolvable.