Normality and dense subspaces

In the first section of this paper, using certain powerful results in $C_{p}$-theory, we show that there exists a nice linear topological space $X$of weight $\omega _{1}$ such that no dense subspace of $X$ is normal. In the second and third sections a natural generalization of normality, called dense normality, is considered. In particular, it is shown in section 2 that the space $R^{c}$ is not normal on some countable dense subspace of it, while it is normal on some other dense subspace. An example of a Tychonoff space $X$, which is not densely normal on a dense separable metrizable subspace, is constructed. In section 3, a link between dense normality and relative countable compactness is established. In section 4 the result of section 1 is extended to densely normal spaces.