Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1049-1061, 2011

# A note on transitively $D$-spaces

## Liang-Xue Peng

Liang-Xue Peng, College of Applied Science, Beijing University of Technology, Beijing 100124, P. R. China, e-mail: pengliangxue@bjut.edu.cn

Abstract: 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 Lindelof 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 Frechet and has a point-countable $wcs^*$-network (or is a hereditarily meta-Lindelof space), then $(X, {\mathcal T})$ is a transitively $D$-space.

Keywords: transitively $D$, sequential, discretely Lindelof, $wcs^*$-network

Classification (MSC 2010): 54F99, 54G99

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