Set-theoretic constructions of nonshrinking open covers

A family {M_α|α?A} is a shrinking of a cover {O_α|α?A} of a topological space if {M_α|α?A} also covers and M_α?O_α for all α?A.?⁺⁺ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.? implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in 17] to show that any space which has a normal product with every P-space is metrizable.