遗传贝尔性质的A-不变性及相关结果
On the $A$-invariance of the hereditary Baire property and related results
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中文总结 AI 辅助
该研究证明了自由阿贝尔拓扑群A(X)与A(Y)拓扑同构时，X和Y的遗传贝尔性质、强σ-分散性/性质(κ)、分散性分别等价，丰富了拓扑群与拓扑空间性质关联的研究。
中文摘要 AI 辅助
我们证明：若X和Y是第一可数完全空间，且自由阿贝尔拓扑群A(X)与A(Y)拓扑同构，则X是遗传贝尔空间当且仅Y也是遗传贝尔空间。我们还证明：对任意吉洪诺夫空间，若存在C_p(X)到C_p(Y)的连续线性满射，且X要么是强σ-分散空间，要么具有性质(κ)，则Y也满足这些性质。此外，我们得到：若X和Y是吉洪诺夫空间，其中每个闭集都有一个W-点，且自由阿贝尔拓扑群A(X)与A(Y)拓扑同构，则X是分散空间当且仅Y是分散空间。
英文摘要
We prove that if $X$ and $Y$ are first-countable perfect spaces such that the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is a hereditarily Baire space if and only if $Y$ is hereditarily Baire as well. We also establish that for any Tychonoff space, if there exists a continuous linear surjection of the space $C_p(X)$ onto the space $C_p(Y)$ and the space $X$ is either strongly $σ$-scattered or has property $(κ)$, then $Y$ also satisfies those properties. Additionally, we obtain the following result: if $X$ and $Y$ are Tychonoff spaces in which every closed set has a $W$-point, and if the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is scattered if and only if $Y$ is scattered.
