Abstract:We shall prove that under CH every regular meta-Lindel\"of $P$-space which is locally $D$ has the $D$-property. In addition, we shall prove that a regular submeta-Lindel\"of $P$-space is {\it D\/} if it is locally $D$ and has locally extent at most~$\omega_1$. Moreover, these results can be extended from the class of locally $D$-spaces to the wider class of $\mathbb D$-scattered spaces. Also, we shall give a direct proof (without using topological games) of the result shown by Peng [{\it On spaces which are D, linearly D and transitively D\/}, Topology Appl. {\bf 157} (2010), 378--384] which states that every weak $\overline{\theta}$-refinable $\mathbb D$-scattered space is~$D$.
Keywords: property $D$; meta-Lindel\"of; weak $\overline{\theta}$-refinable; $P$-space; scattered space
AMS Subject Classification: 54D20 54A35 54G10