A new small Dowker space

TL;DR

The paper demonstrates that under certain set-theoretic assumptions, such as the existence of a Luzin set or specific combinatorial principles, a Dowker space of size 1 can be constructed, linking set theory with topology.

Contribution

It establishes the existence of a Dowker space of size 1 under various set-theoretic hypotheses, providing new connections between set theory and topology.

Findings

Existence of Dowker space of size 1 under Luzin set assumption.

Existence of Dowker space of size 1 under stick or diamond principles.

Strong guessing principles hold at 1 under these hypotheses.

Abstract

It is proved that if there exists a Luzin set, or if either the stick principle or diamond(b) hold, then a strong instance of the guessing principle $\clubsuit_{AD}$ holds at the first uncountable cardinal. In particular, any of the above hypotheses entails the existence of a Dowker space of size $\aleph_1$.