A note on transitively $D$-spaces

Peng, Liang-Xue

Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1049-1061.

Résumé

In this note, we show that if for any transitive neighborhood assignment $\varphi$ for $X$ there is a point-countable refinement $\mathcal{F}$ such that for any non-closed subset $A$ of $X$ there is some $V\in \mathcal{F}$ such that $|V\cap A|\ge \omega$, then $X$ is transitively $D$. As a corollary, if $X$ is a sequential space and has a point-countable $wc{s}^{\ast }$-network then $X$ is transitively $D$, and hence if $X$ is a Hausdorff $k$-space and has a point-countable $k$-network, then $X$ is transitively $D$. We prove that if $X$ is a countably compact sequential space and has a point-countable $wc{s}^{\ast }$-network, then $X$ is compact. We point out that every discretely Lindelöf space is transitively $D$. Let $(X,\tau )$ be a space and let $(X,\mathcal{T})$ be a butterfly space over $(X,\tau )$. If $(X,\tau )$ is Fréchet and has a point-countable $wc{s}^{\ast }$-network (or is a hereditarily meta-Lindelöf space), then $(X,\mathcal{T})$ is a transitively $D$-space.

Détail

DOI : 10.1007/s10587-011-0047-5

Classification : 54D20, 54F99, 54G99
Mots-clés : transitively $D$; sequential; discretely Lindelöf; $wc{s}^{\ast }$-network