积空间的选择覆盖性质
Selective covering properties of product spaces
摘要：
我们研究了选择性覆盖性质的保持，包括经典的Menger，Hurewicz，Rothberger，Gerlits和Nagy，以及其他人，在产品下的一些主要家庭的reals的集中集。我们的方法包括投影方法介绍了作者在早期的工作，以及几个新的方法。一些特殊的后果，我们的主要结果是（定义提供的文件）：\be \item每一个产品的集中空间与Hurewicz $\sone(\Ga,\Op)$空间满足$\sone(\Ga,\Op)$。另一方面，假设\CH{}，对于每个Sierpienski集$S$，存在一个Luzin集$L$，使得$L\x S$可以通过Borel函数映射到实数线上。\item假设半滤子三分法，每个集中空间是有效的Menger和有效的Rothberger。\item每个音阶集都是有效的Hurewicz，有效的Menger，有效的Scheepers，和有效的Gerlits-Nagy. \item，每个有效的Lindelöf空间都是有效的Hurewicz，有效的Menger和有效的Scheepers。\ee一个臭名昭著的公开问题是问Rothberger性质的可加性是否可以严格大于$\add(\cN)$，即实数的Lebesgue-null集的理想的可加性。我们得到了一个肯定的答案，模半过滤器三分法与$\add(\cN)<\cov(\cM)$的一致性。我们的结果改进和统一的一些结果，早期建立的许多作者。
英文摘要：
We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz $\sone(\Ga,\Op)$ space satisfies $\sone(\Ga,\Op)$. On the other hand, assuming \CH{}, for each Sierpiński set $S$ there is a Luzin set $L$ such that $L\x S$ can be mapped onto the real line by a Borel function. \item Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. \item Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits--Nagy. \item Assuming $\fd=\aleph_1$, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers. \ee A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than $\add(\cN)$, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy with $\add(\cN)<\cov(\cM)$. Our results improve upon and unify a number of results, established earlier by many authors.
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