A note on transitively D -spaces

Liang-Xue Peng

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 4, page 1049-1061
  • ISSN: 0011-4642

Abstract

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In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ℱ such that for any non-closed subset A of X there is some V ∈ ℱ such that | V ∩ A | ≥ ω , then X is transitively D . As a corollary, if X is a sequential space and has a point-countable w c s * -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 w c s * -network, then X is compact. We point out that every discretely Lindelöf space is transitively D . Let ( X , τ ) be a space and let ( X , 𝒯 ) be a butterfly space over ( X , τ ) . If ( X , τ ) is Fréchet and has a point-countable w c s * -network (or is a hereditarily meta-Lindelöf space), then ( X , 𝒯 ) is a transitively D -space.

How to cite

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Peng, Liang-Xue. "A note on transitively $D$-spaces." Czechoslovak Mathematical Journal 61.4 (2011): 1049-1061. <http://eudml.org/doc/196597>.

@article{Peng2011,
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|\ge \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.},
author = {Peng, Liang-Xue},
journal = {Czechoslovak Mathematical Journal},
keywords = {transitively $D$; sequential; discretely Lindelöf; $wcs^*$-network; transitively ; sequential; discretely Lindelöf; -network},
language = {eng},
number = {4},
pages = {1049-1061},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on transitively $D$-spaces},
url = {http://eudml.org/doc/196597},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Peng, Liang-Xue
TI - A note on transitively $D$-spaces
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 1049
EP - 1061
AB - 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|\ge \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.
LA - eng
KW - transitively $D$; sequential; discretely Lindelöf; $wcs^*$-network; transitively ; sequential; discretely Lindelöf; -network
UR - http://eudml.org/doc/196597
ER -

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