The Suslinian number and other cardinal invariants of continua
Tom 209 / 2010
Streszczenie
By the Suslinian number $\mathop{\rm Sln}(X)$ of a continuum $X$ we understand the smallest cardinal number $\kappa$ such that $X$ contains no disjoint family $\mathbb C$ of non-degenerate subcontinua of size $|\mathbb C|>\kappa$. For a compact space $X$, $\mathop{\rm Sln}(X)$ 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 $\le\mathop{\rm Sln}(X)^+$ and is the limit of an inverse well-ordered spectrum of length $\le \mathop{\rm Sln}(X)^+$, consisting of compacta with weight $\le\mathop{\rm Sln}(X)$ and monotone bonding maps. Moreover, $w(X)\le\mathop{\rm Sln}(X)$ if no $\mathop{\rm Sln}(X)^+$-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of Daniel et al. [Canad. Math. Bull. 48 (2005)]. 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 $\mathop{\rm Sln}(X)<2^{\aleph_0}$, then $X$ is 1-dimensional, has rim-weight $\le\mathop{\rm Sln}(X)$ and weight $w(X)\ge\mathop{\rm Sln}(X)$. Our main tool is the inequality $w(X)\le\mathop{\rm Sln}(X)\cdot w(f(X))$ holding for any light map $f:X\to Y$.
