Relative normality and dense subspaces

In the first section of this paper, refining and using certain techniques of Cp-theory, we show that there exists a nice linear topological space X of weight ω1 such that no dense subspace of X is normal in X (and in itself). This improves a statement in A.V. Arhangelʹskii [Proc. Amer. Math. Soc. 130 (2001) 283–291]. t countable Tychonoff space without a dense normal subspace we identify in Section 2. Of course, this space is not homeomorphic to a topological group. third section we compare three definitions of relative normality, presenting several examples.