A counterexample related to a theorem of Komjath and Weiss

Abstract

In a paper from 1987, Komjath and Weiss proved that for every regular topological space $X$ of character less than $\mathfrak{b}$, if $X\to (top\omega +1{)}_{\omega }^1$, then $X\to (top\alpha {)}_{\omega }^1$ for all $\alpha <{\omega }_1$. In addition, assuming $♢$, they constructed a space $X$ of size continuum, of character $\mathfrak{b}$, satisfying $X\to (top\omega +1{)}_{\omega }^1$, but not $X\to (top{\omega }^2+1{)}_{\omega }^1$. Here, a counterexample space with the same characteristics is obtained outright in ZFC.