A counterexample related to a theorem of Komjáth and Weiss

Joint work with Rodrigo Rey Carvalho.

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

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