A 0-dimensional, Lindel\'of space that is not strongly D
| Autor: | Soukup, Daniel T., Szeptycki, Paul J. |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove that $\diamondsuit$ implies that there is an $HFC_w$ space in $2^{\omega_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindel\"of) which is not strongly $D$. We also show that any $HFC$ space $X$ is dually discrete and if additionally, countable sets have Menger closure then $X$ is a $D$-space. Comment: 14 pages, 1 figure, sumitted to Topology and its Applications |
| Databáze: | arXiv |
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