On resolvability, connectedness and pseudocompactness

TL;DR

This paper constructs specific topological spaces with properties like pseudocompactness, connectedness, and non-resolvability, demonstrating new relationships between these properties under various set-theoretic assumptions.

Contribution

It provides new examples of topological spaces with prescribed properties, advancing understanding of resolvability, connectedness, and pseudocompactness in topology.

Findings

Existence of submaximal dense subspaces in certain powers of $T_1$ spaces.

Construction of pseudocompact, connected, non-$oldsymbol{ ext{kappa}^+}$-resolvable spaces.

Spaces where all continuous real-valued functions are constant, yet they are pseudocompact and connected.

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 $\frak{c}\leq\kappa=\kappa^\omega<\lambda$ 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.