Normality and dense subspaces (Q2750916)

A Tikhonov space \(X\) is said to be normal on its subspace \(Y\) if every two disjoint closed subsets \(A,B\) of \(X\) with \(A\subset \overline{A\cap Y}, B\subset\overline{B\cap Y}\), are contained in two disjoint open subsets of \(X\); \(X\) is said to be densely normal if \(X\) is normal on a dense subspace of \(X\). Some results: 1.1: \(C_p(\omega_1+1)\) does not contain a dense normal subspace; moreover, (4.2), no dense subspace of \(C_p(\omega_1+1)\) is densely normal -- instead of \(C_p(\omega_1+1)\) one may take a submetrizable space with ccc (thus realcompact). 2.7: There exists a countable dense subspace \(Y\) of \(\mathbb{R}^{2^\omega}\) such that \(\mathbb{R}^{2^\omega}\) is not normal on \(Y\). 3.1: If \(X\) is a locally connected pseudocompact space that is normal on its dense subspace \(Y\), then \(Y\) is countably compact in \(X\).