A connected pseudocompact space

Abstract

We construct, under the continuum hypothesis, a first countable connected pseudocompact Tychonoff space without a dense relatively countably compact subset. We do this by blending an Ostaszewski-type construction with a Stephenson-type Ψ-space construction. We also show that any locally pseudocompact first countable regular space with a dense locally conditionally compact subset can be embedded in a pseudocompact first countable regular space. This sheds light on Stephenson's problem whether any locally pseudocompact first countable regular space can be embedded in a pseudocompact first countable regular space.