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- W2897225476 abstract "By a ballean we understand a set $X$ endowed with a family of entourages which is a base of some coarse structure on $X$. Given two unbounded ballean $X,Y$ with normal product $Xtimes Y$, we prove that the balleans $X,Y$ have bounded growth and the bornology of $Xtimes Y$ has a linearly ordered base. A ballean $(X,mathcal E_X)$ is defined to have bounded growth if there exists a function $G$ assigning to each point $xin X$ a bounded subset $G[x]subset X$ so that for any bounded set $Bsubset X$ the union $bigcup_{xin B}G[x]$ is bounded and for any entourage $Einmathcal E_X$ there exists a bounded set $Bsubset X$ such that $E[x]subset G[x]$ for all $xin Xsetminus B$. We prove that the product $Xtimes Y$ of two balleans has bounded growth if and only if $X$ and $Y$ have bounded growth and the bornology of the product $Xtimes Y$ has a linearly ordered base. Also we prove that a ballean $X$ has bounded growth (and the bornology of $X$ has a linearly ordered base) if its symmetric square $[X]^{le 2}$ is normal (and the ballean $X$ is not ultranormal). A ballean $X$ has bounded growth and its bornology has a linearly ordered base if for some $nge 3$ and some subgroup $Gsubset S_n$ the $G$-symmetric $n$-th power $[X]^n_G$ of $X$ is normal. On the other hand, we prove that for any ultranormal discrete ballean $X$ and every $nge 2$ the power $X^n$ is not normal but the hypersymmetric power $[X]^{le n}$ of $X$ is normal. Also we prove that the finitary ballean of a group is normal if and only if it has bounded growth if and only if the group is countable." @default.
- W2897225476 created "2018-10-26" @default.
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- W2897225476 date "2018-10-18" @default.
- W2897225476 modified "2023-09-27" @default.
- W2897225476 title "The normality and bounded growth of balleans" @default.
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