# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## The Suslinian number and other cardinal invariants of continua

### Tom 209 / 2010

Fundamenta Mathematicae 209 (2010), 43-57 MSC: Primary 54F15; Secondary 54C05, 54F05, 54F50. DOI: 10.4064/fm209-1-4

#### 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 $f:X\to Y$.

#### Autorzy

• T. BanakhUniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego
Kielce, Poland
and
Department of Mathematics
Ivan Franko Lviv National University
Lviv, Ukraine
e-mail
• V. V. FedorchukFaculty of Mechanics and Mathematics
Lomonosov Moscow State University
Vorob'evy Gory, 1
Moscow, Russia
e-mail
• J. NikielInstytut Matematyki i Informatyki
Uniwersytet Opolski
Oleska 48
45-052 Opole, Poland
e-mail
• M. TuncaliNipissing University