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Keywords:
property $D$; meta-Lindelöf; weak $\overline{\theta}$-refinable; $P$-space; scattered space
property $D$; meta-Lindelöf; weak $\overline{\theta}$-refinable; $P$-space; scattered space
Summary:
We shall prove that under CH every regular meta-Lindelöf $P$-space which is locally $D$ has the $D$-property. In addition, we shall prove that a regular submeta-Lindelöf $P$-space is $D$ 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 [On spaces which are D, linearly D and transitively D, Topology Appl. 157 (2010), 378--384] which states that every weak $\overline{\theta}$-refinable $\mathbb D$-scattered space is $D$.
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