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Keywords:
transitively $D$; sequential; discretely Lindelöf; $wcs^*$-network
Summary:
In this note, we show that if for any transitive neighborhood assignment $\phi$ 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 $|V\cap A|\geq \omega$, then $X$ is transitively $D$. As a corollary, if $X$ is a sequential space and has a point-countable $wcs^*$-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 $wcs^*$-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 $wcs^*$-network (or is a hereditarily meta-Lindelöf space), then $(X, {\mathcal T})$ is a transitively $D$-space.
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