连续统的Suslinian数和其他基数不变量
The Suslinian number and other cardinal invariants of continua
摘要：
通过连续统$X$的{ \em Suslinian数} $\Sln(X)$，我们理解最小基数$κ$，使得$X$不包含大小为$|\C|\geκ$的非退化子连续统的不相交族$\C$。对于紧致空间$X$，$\Sln(X)$是包含$X$的同胚副本的连续统的最小Suslinian数。我们的主要结果断言，每个紧空间$X$有重量$\le\Sln(X)^+$，是长度为$\le \Sln(X)^+$的逆良序谱的极限，由带重量$\le\Sln(X)$的矩阵和单调键映射组成。此外，如果不存在$\Sln(X)^+$ -Suslin树，则为$w(X)\le\Sln(X)$。这意味着在Suslin假设下，所有Suslinian连续统都是可度量化的，这回答了\cite{DNTTT1}的一个问题。另一方面，苏斯林假设的否定等价于存在一个遗传可分的不可度量化的苏斯林连续统。如果$X$是一个连续体与$\Sln(X)<2^{\aleph_0}$，那么$X$是1维的，有边缘权重$\le\Sln(X)$和权重$w(X)\ge\Sln(X)$。我们的主要工具是不等式$w(X)\le\Sln(X)\cdot w(f(X))$，它适用于任何光照贴图$f:X\to Y$。
英文摘要：
By the {\em Suslinian number} $\Sln(X)$ of a continuum $X$ we understand the smallest cardinal number $κ$ such that $X$ contains no disjoint family $\C$ of non-degenerate subcontinua of size $|\C|\geκ$. For a compact space $X$, $\Sln(X)$ is the smallest Suslinian number of a continuum which contains a homeomorphic copy of $X$. Our principal result asserts that each compact space $X$ has weight $\le\Sln(X)^+$ and is the limit of an inverse well-ordered spectrum of length $\le \Sln(X)^+$, consisting of compacta with weight $\le\Sln(X)$ and monotone bonding maps. Moreover, $w(X)\le\Sln(X)$ if no $\Sln(X)^+$-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of \cite{DNTTT1}. On the other hand, the negation of the Suslin Hypothesis is equivalent to the existence of a hereditarily separable non-metrizable Suslinian continuum. If $X$ is a continuum with $\Sln(X)<2^{\aleph_0}$, then $X$ is 1-dimensional, has rim-weight $\le\Sln(X)$ and weight $w(X)\ge\Sln(X)$. Our main tool is the inequality $w(X)\le\Sln(X)\cdot w(f(X))$ holding for any light map $f:X\to Y$.
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