Spaces with a property related to uniformly local finiteness
Spaces with a property related to uniformly local finiteness
复制标题
具有与均匀局部有限性相关的属性的空间
DOI：
10.21099/tkbjm/1496159449
复制
发表时间：
1982
影响因子：
0.7
通讯作者：
T. Hoshina
中科院分区：
文献类型：
--
作者：
T. Hoshina
$X$是一致局部有限的，则称$X$具有性质$(U)$。在K. Morita[10]中定义了这些概念，并指出了每一个m空间或每一个强正规（$=集合正规可数仿紧）空间都是一个具有性质$(U)$的空间，并且这样的空间在L. L. Krajewski[7]意义上是可扩展的。因此，对于正规空间的性质$(U)$，可扩展性和强正态性都是由M. Kat\v{e}tov[6]的一个著名定理相吻合的，因此Morita[10]提出了一个问题，求出一个与可扩展性等价于性质$(U)$的条件。本文的目的是研究具有属性$(U)$的空间，主要是通过定义一个新的U嵌入概念，它是p嵌入的推广；如果$ a $的子集的每一个一致局部有限的集合也在$X$中是一致局部有限的，则空间$X$中的子空间$ a $被称为U嵌入在$X$中。在\S 1中，我们处理具有子集的每一个离散集合是一致局部有限的性质的空间，我们称之为具有性质$(U)^{*}$的空间。根据C. H. Dowker的定义，集合正规空间就是其闭集任意一个都是p嵌入的空间。在此结果的激励下，我们将建立一个定理，证明一个空间$X$具有性质$(U)^{*}$，如果$X$的任何闭集$X$是U嵌入到$X$中，然后证明一个空间具有性质$(U)$，如果它具有性质$(U)^{*}$，并且是J. Mack[9]意义上的bc空间；后者非常类似于Krajewski的一个定理，即空间是可扩展的，如果它是离散可扩展的并且是可数的准紧的。在S 2中，我们将定义具有弱性质的空间$(U)$，它包括所有M ' -空间[5]和所有极端不连通空间，从而给出具有性质的空间$(U)$的另一种描述，这是对上面Morita问题的回答。
$X$ is uniformly locally finite, then $X$ is said to have property $(U)$ . These notions are defined in K. Morita [10], and it is pointed out there that every M-space or every strongly normal ( $=collectionwise$ normal and countably paracompact) space is a space with property $(U)$ , and such a space is expandable in the sense of L. L. Krajewski [7]. Hence for normal spaces property $(U)$ , expandability and strong normality all coincide with each other by a well-known theorem of M. Kat\v{e}tov [6], and so a question was posed by Morita [10] to find a condition which, together with expandability, is equivalent to property $(U)$ . The purpose of this paper is to investigate spaces with property $(U)$ , mainly by defining a new notion of U-embedding which is a generalization of P-embedding; a subspace $A$ of a space $X$ is said to be U-embedded in $X$ if every uniformly locally finite collection of subsets of $A$ is uniformly locally finite also in $X$ In \S 1 we treat spaces having a property that every discrete collection of subsets is uniformly locally finite, which we call spaces with property $(U)^{*}$ . By C. H. Dowker [1], collectionwise normal spaces are precisely those spaces any of whose closed set is P-embedded. Being motivated with this result we shall establish a theorem that a space $X$ has property $(U)^{*}$ iff any closed set of $X$ is U-embedded in $X$, and then it will be shown that a space has property $(U)$ iff it has property $(U)^{*}$ and is a cb-space in the sense of J. Mack [9]; the latter is a quite analogue to a theorem of Krajewski [7] that a space is expandable iff it is discretely expandable and countably paracompact. In \S 2 we shall give another description of spaces with property $(U)$ , which is an answer to the question of Morita above, by defining spaces with weak property $(U)$ that include all M’-spaces [5] and all extremally disconnected spaces.