Shrinking in perfect preimages of shrinking spaces (Q1924680)

An open cover \(\{V_\alpha:\alpha\in k\}\) of a space \(X\) is a shrinking for another open cover \(\{U_\alpha : \alpha\in k\}\) if \(\text{cl}(V_\alpha)\subset U_\alpha\) for every \(\alpha\in k\). A space \(X\) is \(k\)-shrinking if every open cover of cardinality \(\leq k\) has a shrinking; \(X\) is shrinking if it is \(k\)-shrinking for every \(k\). It is shown in the first part of the paper that a \(k\)-paracompact perfect preimage of a shrinking space is \(k^+\)-shrinking. As a corollary the author obtained that a normal perfect preimage of a shrinking space is \(\omega_1\)-shrinking. The aim of the next part is to prove that a normal perfect preimage of a generalized ordered space is shrinking. This result is used then to show that a normal perfect preimage of a monotonically normal space is shrinking.