c.c.c.中的可分离性泛型扩张
Resolvability in c.c.c. generic extensions
摘要：
每个拥挤空间$X$在地面模型的c.c. c通用扩展$V^{Fn(|X|,2})$中是$ω$ -可分解的。我们调查我们可以说$λ$ -resolvability在c.c. c-通用扩展$λ>ω$？一个拓扑空间是“单调$ω_1$ -可分解的”，如果存在一个函数$f:X\to {ω_1}$使得对于每个$α<{ω_1}$有$$\{x\in X: f(x)\ge α \}\subset^{dense}X $$.我们证明了给定一个$T_1$空间$X$，下列陈述是等价的：（1）$X$在某些c.c. c-类属扩张中是$ω_1$ -可分解的，（2）$X$是单调$ω_1$ -可分解的. (3)$X$在Cohen通用扩展$V^{Fn({ω_1},2)}$中是$ω_1$可解析的。我们研究哪些空间是单调PDMATHAMR-可分解的。证明了若拓扑空间$X$是c.c.c，且$ω_1\le Δ(X)\le |X|<ω_ω$，则$X$是单调$ω_1$ -可分解的.另一方面，它也是一致的，模存在的可测基数，有一个空间$Y$与$|Y|=Δ(Y)=\aleph_ω$这是不是单调$ω_1$ -可分解的。在c.c. c类属扩张中${ω_1}$ -可分解性的刻画提出了以下问题：从基模型出发的拥挤空间在$V^{Fn(ω,2)}$中是$ω$ -可分解的吗？我们证明了（i）如果$V=L$，则每个拥挤的c.c.c.空间$X$在$V^{Fn(ω,2)}$中是$ω$ -可分解，（ii）如果不存在弱不可达基数，则每个拥挤空间$X$在$V^{Fn(ω_1,2)}$中是$ω$ -可分解的。另一方面，它也是一致的，有一个拥挤的空间$X$与$|X|=Δ(X)={ω_1}$，使$X$仍然无法解决后，增加一个单一的科恩实。
英文摘要：
Every crowded space $X$ is $ω$-resolvable in the c.c.c generic extension $V^{Fn(|X|,2})$ of the ground model. We investigate what we can say about $λ$-resolvability in c.c.c-generic extensions for $λ>ω$? A topological space is "monotonically $ω_1$-resolvable" if there is a function $f:X\to {ω_1}$ such that $$\{x\in X: f(x)\ge α \}\subset^{dense}X $$ for each $α<{ω_1}$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is $ω_1$-resolvable in some c.c.c-generic extension, (2) $X$ is monotonically $ω_1$-resolvable. (3) $X$ is $ω_1$-resolvable in the Cohen-generic extension $V^{Fn({ω_1},2)}$. We investigate which spaces are monotonically $ω_1$-resolvable. We show that if a topological space $X$ is c.c.c, and $ω_1\le Δ(X)\le |X|<ω_ω$, then $X$ is monotonically $ω_1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=Δ(Y)=\aleph_ω$ which is not monotonically $ω_1$-resolvable. The characterization of ${ω_1}$-resolvability in c.c.c generic extension raises the following question: is it true that crowded spaces from the ground model are $ω$-resolvable in $V^{Fn(ω,2)}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is $ω$-resolvable in $V^{Fn(ω,2)}$, (ii) if there is no weakly inaccesssible cardinals, then every crowded space $X$ is $ω$-resolvable in $V^{Fn(ω_1,2)}$. On the other hand, it is also consistent that there is a crowded space $X$ with $|X|=Δ(X)={ω_1}$ such that $X$ remains irresolvable after adding a single Cohen real.
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