Productivity of paracompactness and closed images of real GO-spaces (Q524350)

Recall Telgársky's game \(G(\mathbf{DC},X)\) involving two players choosing, respectively, odd- and even-indexed closed sets \(E_n\subset X\), \(X\) a topological space, so that \(E_0=X\), \(E_{2n+1}\) lies in \(E_{2n}\) and is the union of a discrete family of compacta, and \(E_{2n+2}\subset E_{2n}\setminus E_{2n+1}\). The first player wins if \(\cap_nE_{2n}=\varnothing\). The present paper considers the converse of Telgársky's theorem for paracompact \(X\): if the first player has a winning strategy in the game \(G(\mathbf{DC},X)\) then \(X\) is productively paracompact. The converse is shown to hold in the following contexts: there are a real GO-space \(G\) and a closed surjective map \(f:G\to X\); there are a GO-space \(G\) defined on a subspace of \(\mathbb R\) and a closed surjective map \(f:G\to X\) such that \(f(L_G)\) is countable, where \(L_G=\{x\in G \mid (\leftarrow,x] \text{ but not } \{x\} \text{ is open in } G\}\).