A characterization of absolute neighborhood retracts
A characterization of absolute neighborhood retracts
复制标题
绝对邻域收缩的表征
DOI：
10.1090/s0002-9904-1942-07652-x
复制
发表时间：
1942
影响因子：
1.3
通讯作者：
R. Fox
中科院分区：
文献类型：
--
作者：
R. Fox
绝对邻域缩回（ANR）我指的是一个可分离的可度量空间它是包含它的所有可分离的可度量空间的邻域缩回，并且它在其中是封闭的。这种对Borsuk的原始定义的推广是由Kuratowski给出的，目的是扩大绝对邻域收缩的类别，使其包括某些不紧的空间。最初由Borsuk指定为绝对邻域收缩（或$K-sets）的空间现在被称为紧凑型绝对邻域收缩。紧反反集的许多性质对更一般的反反集同样成立。Hubert平行四边形Q，即闭合单位区间[0,1]与自身可数次的乘积，是一个“普适”紧ANR，因为每个紧ANR都与Q的一个邻域缩回同纯。Borsuk的经典理论很好地利用了紧ANR集在Q中的嵌入，这里解决的问题是寻找一个“普适”ANR。
By an absolute neighborhood retract (ANR) I mean a separable metrizable space which is a neighborhood retract of every separable metrizable space which contains it and in which it is closed. This generalization of Borsuk's original definition was given by Kuratowski for the purpose of enlarging the class of absolute neighborhood retracts to include certain spaces which are not compact. The space originally designated by Borsuk as absolute neighborhood retracts (or $K-sets) will now be referred to as compact absolute neighborhood retracts. Many of the properties of compact ANR-sets hold equally for the more general ANR-sets. The Hubert parallelotope Q, that is, the product of the closed unit interval [0, 1 ] with itself a countable number of times is a "universal" compact ANR in the sense that every compact ANR is homeomorphic to a neighborhood retract of Q. The classical theory of Borsuk makes good use of the imbedding of compact ANR-sets in Q. The problem solved here is that of finding a "universal" ANR.