拓扑积中的Noether型
Noetherian type in topological products
摘要：
拓扑空间$X$（Nt（X））的基数不变量“诺特型”由Peregudov在1997年引入，以处理早在1976年就由俄罗斯学派研究的基本性质。我们研究了它在拓扑空间的乘积和盒积中的行为。我们在第2节中证明了：1）存在空间$X$和$Y$使得$Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$。2)在几类紧空间中，诺特型通过形成平方和传递到稠密子空间的操作而保持。在第3节中，给出了具有可数盒拓扑$(2^{\aleph_ω})_δ$的权$\aleph_ω$的康托立方体的诺特型，它与$\aleph_ω$的可数子集的覆盖集合的组合学密切相关。我们讨论了像$\square_{\aleph_ω}$和张的猜想$\aleph_ω$的原则对这个数的影响，并证明了它是不可判定的ZFC（相对于ZFC与大基数公理的一致性）。在PCF理论中，我们建立了$\aleph_ω$的可数子集的$(\aleph_4,\aleph_1)$ -稀疏覆盖族。由此得出$(2^{\aleph_ω})_δ$的Noether型上的$\aleph_4$的绝对上界。证明使用的想法从希拉的证明，如果$κ^+ <λ$，那么他的理想$I[λ]$包含一个固定的集合组成的点的共尾$κ$。
英文摘要：
The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_ω$ with the countable box topology, $(2^{\aleph_ω})_δ$, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of $\aleph_ω$. We discuss the influence of principles like $\square_{\aleph_ω}$ and Chang's conjecture for $\aleph_ω$ on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_ω$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_ω})_δ$. The proof uses ideas from Shelah's proof that if $κ^+ <λ$ then his ideal $I[λ]$ contains a stationary set consisting of points of cofinality $κ$.
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