Abstract:We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space , the function space is not countable dense homogeneous. This answers a question posed recently by R. Hernández-Gutiérrez. We also conclude that, for any infinite-dimensional Banach space (dual Banach space ), the space equipped with the weak topology ( with the weak topology) is not countable dense homogeneous. We generalize some results of Hrušák, Zamora Avilés, and Hernández-Gutiérrez concerning countable dense homogeneous products.

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中文翻译：

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一个可数的密集齐次拓扑向量空间是一个Baire空间

摘要：我们证明，每个包含Cantor集副本的齐次可数稠密齐次拓扑空间都是Baire空间。特别地，每个可数的密集均质拓扑向量空间都是一个Baire空间。由此可见，对于任何非离散的可度量空间，函数空间都不是可计数的密集齐次的。这回答了赫尔南德斯-古铁雷斯最近提出的一个问题。我们还得出结论，对于任何无限维Banach空间（双重Banach空间），配备有弱拓扑（具有弱拓扑）的空间都是不可数的密集同质。我们归纳了Hrušák，ZamoraAvilés和Hernández-Gutiérrez关于可数致密均质产品的一些结果。

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