On the absolutes of compact spaces with a minimally acting group
On the absolutes of compact spaces with a minimally acting group
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关于具有最小作用群的紧凑空间的绝对性
DOI：
10.1090/s0002-9939-1994-1246512-x
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发表时间：
1994
影响因子：
1.3
通讯作者：
I. Bandlow
中科院分区：
文献类型：
--
作者：
I. Bandlow
如果 w 有界群 G 连续作用于紧豪斯多夫空间 X 且 X 中每个点的轨道都是稠密的，则 X 绝对是康托立方。如果 G 是 X 的同胚群，并且自然映射 X x G —> X 是连续的，则拓扑群 G 连续作用于拓扑空间 X。 X 始终假设为紧豪斯多夫空间。我们说，如果轨道 {g(x): g £ G} 在 X 中对于每个点 x £ X 都是稠密的，则 G 对 X 的作用最小。很容易看出，如果对于 X 的每个非空开子集 U，存在 gx, ... , g„ £ G 使得 gx(U)U---Ug„(U) = X，则 G 对 X 的作用最小。Balear 和 Blaszczyk 在 [2] 中证明，每个具有最小作用可数群的零维紧空间是如果两个紧致空间各自的正则开子集的布尔代数是同构的，那么根据 Uspenskij [14] 和 Shapiro [12] 的结果，如果 («-有界群 G 传递且连续地作用于紧致空间 X，则 X 与康托尔立方体是同绝对的。当且仅当对于任何邻域 U 时，拓扑群 G 才被称为 «-有界的。其中性元素存在 G 的可数子集 A，使得 G = AU（Guran，参见 Archangelskij [1]）。本文的目的是证明以下定理 1。如果共界群 G 连续且最小地作用于紧致 Hausdorff 空间 X，则 X 与 Cantor 立方体共绝对。该证明基于 Shapiro [12, 13] 的结果，并利用了 Uspenskij [14] 和Balear 和 Blaszczyk [2] 首先，我们需要 Shapiro 的以下杰出结果： (1) 对于每个非空开子集 U，如果 w(U) = w(X)，则每个关于权值齐次的二进紧空间都与康托立方体同绝对。 1991 年 6 月 23 日编辑收到。初级 54D80、22A05。
If an w-bounded group G acts continuously on a compact Hausdorff space X and the orbit of every point is dense in X , then X iscoabsolute to a Cantor cube. A topological group G acts continuously on the topological space X if G is a group of homeomorphisms of X and the natural map X x G —> X is continuous. X is always assumed to be a compact Hausdorff space. We say that G acts minimally on X if the orbit {g(x): g £ G} is dense in X for every point x £ X. It is easy to see that G acts minimally on X if for every nonempty open subset U of X, there exist gx, ... , g„ £ G such that gx(U)U---Ug„(U) = X. Balear and Blaszczyk proved in [2] that every zero-dimensional compact space with a minimally acting countable group is coabsolute to a Cantor cube. Two compact spaces are said to be coabsolute if their respective Boolean algebras of regular open subsets are isomorphic. From results of Uspenskij [14] and Shapiro [12] it follows that if an («-bounded group G acts transitively and continuously on a compact space X, then X is coabsolute to a Cantor cube. A topological group G is said to be «-bounded if and only if for any neighbourhood U of its neutral element there is a countable subset A of G such that G = AU (Guran, see Archangelskij [1]). The aim of this note is to prove the following Theorem 1. If an co-bounded group G acts continuously and minimally on the compact Hausdorff space X, then X is coabsolute to a Cantor cube. The proof is based on results of Shapiro [12, 13] and makes use of arguments of Uspenskij [14] and Balear and Blaszczyk [2]. At first, we need the following outstanding result of Shapiro: (1) Every dyadic compact space which is homogeneous with respect to the weight is coabsolute to a Cantor cube. A space X is said to be homogeneous with respect to the weight if w(U) = w(X) for every nonempty open subset U. Received by the editors June 23, 1992. 1991 Mathematics Subject Classification. Primary 54D80, 22A05.