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Register a new userThe Suslinian number and other cardinal invariants of continua
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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 $|C|gekappa$. 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 $leSln(X)^+$ and is the limit of an inverse well-ordered spectrum of length $le Sln(X)^+$, consisting of compacta with weight $leSln(X)$ and monotone bonding maps. Moreover, $w(X)leSln(X)$ 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 $leSln(X)$ and weight $w(X)geSln(X)$. Our main tool is the inequality $w(X)leSln(X)cdot w(f(X))$ holding for any light map $f:Xto Y$.
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Definition. Let $kappa$ be an infinite cardinal, let {X(i)} be a (not necessarily faithfully indexed) set of topological spaces, and let X be the product of the spaces X(i). The $kappa$-box product topology on X is the topology generated by those products of sets U(i) for which (a) for each i, U(i) is open in X(i); and (b) U(i) = X(i) with fewer than $kappa$-many exceptions. (Thus, the usual Tychonoff product topology on X is the $omega$-box topology.) With emphasis on weight, density character, and Souslin number, the authors study and determine the value of several cardinal invariants on the space X with its $kappa$-box topology, in terms of the corresponding invariants of the individual spaces X(i). To the authors knowledge, this work is the first systematic study of its kind. Some of the results are axiom-sensitive, and some duplicate (and extend, and make precise) earlier work of Hewitt-Marczewski-Pondiczery, of Englking-Karlowicz, of Comfort-Negrepontis, and of Cater-Erdos-Galvin.
A function $f:Xto Y$ between topological spaces is called $sigma$-$continuous$ (resp. $barsigma$-$continuous$) if there exists a (closed) cover ${X_n}_{ninomega}$ of $X$ such that for every $ninomega$ the restriction $f{restriction}X_n$ is continuous. By $mathfrak c_sigma$ (resp. $mathfrak c_{barsigma}$) we denote the largest cardinal $kappalemathfrak c$ such that every function $f:Xtomathbb R$ defined on a subset $Xsubsetmathbb R$ of cardinality $|X|<kappa$ is $sigma$-continuous (resp. $barsigma$-continuous). It is clear that $omega_1lemathfrak c_{barsigma}lemathfrak c_sigmalemathfrak c$. We prove that $mathfrak plemathfrak q_0=mathfrak c_{barsigma}=min{mathfrak c_sigma,mathfrak b,mathfrak q}lemathfrak c_sigmalemin{mathrm{non}(mathcal M),mathrm{non}(mathcal N)}$. The equality $mathfrak c_{barsigma}=mathfrak q_0$ resolves a problem from the initial version of the paper.
We introduce and study oscillator topologies on paratopological groups and define certain related number invariants. As an application we prove that a Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology provided $G$ is 3-oscillating. A paratopological group $G$ is 3-oscillating (resp. 2-oscillating) provided for any neighborhood $U$ of the unity $e$ of $G$ there is a neighborhood $Vsubset G$ of $e$ such that $V^{-1}VV^{-1}subset UU^{-1}U$ (resp. $V^{-1}Vsubset UU^{-1}$). The class of 2-oscillating paratopological groups includes all collapsing, all nilpotent paratopological groups, all paratopological groups satisfying a positive law, all paratopological SIN-group and all saturated paratopological groups (the latter means that for any nonempty open set $Usubset G$ the set $U^{-1}$ has nonempty interior). We prove that each totally bounded paratopological group $G$ is countably cellular; moreover, every cardinal of uncountable cofinality is a precaliber of $G$. Also we give an example of a saturated paratopological group which is not isomorphic to its mirror paratopological group as well as an example of a 2-oscillating paratopological group whose mirror paratopological group is not 2-oscillating.
We show that if $G$ is a finitely generated group of subexponential growth and $X$ is a Suslinian continuum, then any action of $G$ on $X$ cannot be expansive.
If $f:[a,b]to mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:mathbb{C}tomathbb{C}$ is map and $X$ is a continuum. We extend the above for certain continuous maps of dendrites $Xto D, Xsubset D$ and for positively oriented maps $f:Xto mathbb{C}, Xsubset mathbb{C}$ with the continuum $X$ not necessarily invariant. Then we show that in certain cases a holomorphic map $f:mathbb{C}tomathbb{C}$ must have a fixed point $a$ in a continuum $X$ so that either $ain mathrm{Int}(X)$ or $f$ exhibits rotation at $a$.
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