Lifting the Collins-Roscoe property by condensations (Q2850634)

If there exists a condensation (a continuous bijection) from a compact space \(X\) onto a space with a topological property \(\mathcal{P}\), then, since the map is a homeomorphism, \(X\) has \(\mathcal{P}\). In this paper, the author presents results of this type for the class of Lindelöf \(\Sigma\)-spaces which contains the class of compact spaces. Indeed, he shows that if a Lindelöf \(\Sigma\)-space \(X\) condenses onto a monotonically monolithic space (a space with the Collins-Roscoe property), then \(X\) is monotonically monolithic (has the Collins-Roscoe property). Furthermore, he gives the following results with respect to monotonic monolithicity and the Collins-Roscoe property: if \(X\) is a monotonically \(\omega\)-monolithic perfect normal compact space, then \(X\) is metrizable; if \(X\) is a strongly monotonically monolithic Lindelöf \(\Sigma\)-space, then \(X\) is second countable; if a Lindelöf \(\Sigma\)-space \(X\) condenses into a \(\Sigma_s\)-product of spaces of countable \(i\)-weight, then \(X\) has the Collins-Roscoe property. He also gives a negative answer to a question of \textit{G. Gruenhage} in [Topology Appl. 159, No. 6, 1559--1564 (2012; Zbl 1241.54013)], that is, he proves that there exists a compact space which is not Gul'ko compact but has the Collins-Roscoe property.