Weakly Corson compact trees (Q2121635)

A tree \((T, \leq)\) is called rooted if it has a unique minimal element \(r_T\) called root. If \(t\in T\), then \(V_t=\{s\in T: t\leq s\}\) and \(I(T)=\{t\in T: \) either \(t=r_T\) or \(t\) has a predecessor\(\}\). A set \(A\subset T\) is a chain if \((A,\leq)\) is a linearly ordered set. The tree \(T\) is chain complete, if every chain in \(T\) has a supremum. The coarse wedge topology \(\tau_{cw}\) on a tree \(T\) is generated by the subbase \(\{V_t, T\setminus V_t: t\in I(T)\}\). A compact space \(K\) is Corson compact if it is embeddable in the \(\Sigma\)-product \(\Sigma(A)= \{x\in \mathbb R^A: |x^{-1}(\mathbb R\setminus\{0\})| \leq\omega\}\) for some set \(A\). A compact space \(K \subset \mathbb R^A\) is Valdivia compact if \(K \cap \Sigma(A)\) is dense in \(K\). Furthermore, \(K\) is weakly Valdivia compact if it is a continuous image of a Valdivia compact space. A compact space \(K\) is called weakly Corson compact if \(K\) is a continuous image of a countably compact subspace of \(\Sigma(A)\) for some set \(A\). The authors prove, among other things, that, for any rooted chain complete tree \(T\), if \((T, \tau_{cw})\) is a weakly Corson compact space, then it has a dense subset of \(G_\delta\)-points. Consider the set \(T_\alpha=\{0,1\}^\alpha\) for each \(\alpha\leq\omega_1\) and let \(T=\bigcup\{T_\alpha: \alpha \leq \omega_1\}\). For any \(f,g\in T\), it is declared that \(f\leq g\) if and only if \(f\subset g\). Then \((T, \leq)\) is a tree such that \((T, \tau_{cw})\) is a compact Hausdorff space. The main result of the paper states that \((T,\tau_{cw})\) is not weakly Corson compact but every closed subspace of \((T,\tau_{cw})\) is weakly Valdivia compact.