Topological Dynamics indexed by words

Infinitary Noetherian Constructions II. Transfinite Words and the Regular Subword Topology

We show that the spaces of transfinite words, namely ordinal-indexed words, over a Noetherian space, is also Noetherian, under a natural topology which we call the regular subword topology. We characterize its sobrification and its specialization ordering, and we give an upper bound on its dimension and on its stature.

On topological groups admitting a base at identity indexed with $ω^ω$

A topological group $G$ is said to have a local $ω^ω$-base if the neighbourhood system at identity admits a monotone cofinal map from the directed set $ω^ω$. In particular, every metrizable group is such, but the class of groups with a local $ω^ω$-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-arichimedean ordered fields lead to natural families of non-metrizable groups with a local $ω^ω$-base which nevertheless are Baire topological spaces. More examples come from…

Cardinal-indexed classifying spaces for families of subgroups of any topological group

For $G$ a topological group, existence theorems by Milnor (1956), Gelfand-Fuks (1968), and Segal (1975) of classifying spaces for principal $G$-bundles are generalized to $G$-spaces with torsion. Namely, any $G$-space approximately covered by tubes (a generalization of local trivialization) is the pullback of a universal space indexed by the orbit types of tubes and cardinality of the cover. For $G$ a Lie group, via a metric model we generalize the corresponding uniqueness theorem by Palais (1960) and Bredon (1972) for compact $G$. Namely, the $G$-homeomorphism types of proper $G$-spaces over …

由偏序集索引的Stone空间划分

Stone space partitions indexed by a poset

本文研究由偏序集或PO系统索引的Stone空间修剪划分理论，分析序性质与拓扑性质的关联、划分的扩张完备性、存在充要条件及同胚唯一性，得到Cantor集上满足特定闭包条件的实索引划分推论。

Central sets generated by uniformly recurrent words

A subset $A$ of $\nats$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \nats} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing rich combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. In this paper we investigate central sets in the framework of combinatorics on words. Using various families of uniformly r…