The Suslinian number and other cardinal invariants of continua

T. Banakh, V. V. Fedorchuk, J. Nikiel, M. Tuncali

Published 2009-03-04Version 2

By the {\em Suslinian number} $\Sln(X)$ of a continuum $X$ we understand the smallest cardinal number $\kappa$ such that $X$ contains no disjoint family $\C$ of non-degenerate subcontinua of size $|\C|\ge\kappa$. 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 $f:X\to Y$.