A connected \(F\)-space (Q850596)

\textit{Y. A. Abramovich} and \textit{A. K. Kitover} asked [Positivity 7, No.1-2, 95--97 (2003; Zbl 1046.46005)] whether maximal \(d\)-independent systems in vector lattices are \(d\)-bases. To answer the problem in the negative, it suffices to find a compact and connected \(F\)-space \(X\) and a continuous function \(f:X\to \mathbb R\) such that union of interiors of all \(f^{-1}(x)\) is not dense in \(X\). Such space is constructed in the present paper -- it is a subcontinuum of the Čech--Stone compactification of \(\omega\times [0,1]^2\).