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Tukey order and diversity of free Abelian topological groups
Journal of Pure and Applied Algebra ( IF 0.7 ) Pub Date : 2021-02-15 , DOI: 10.1016/j.jpaa.2021.106712
Paul Gartside

For a Tychonoff space X the free Abelian topological group over X, denoted A(X), is the free Abelian group on the set X with the coarsest topology so that for any continuous map of X into an Abelian topological group its canonical extension to a homomorphism on A(X) is continuous.

We show there is a family A of maximal size, 2c, consisting of separable metrizable spaces, such that if M and N are distinct members of A then A(M) and A(N) are not topologically isomorphic (moreover, A(M) neither embeds topologically in A(N) nor is an open image of A(N)). We show there is a chain C={Mα:α<c+}, of maximal size, of separable metrizable spaces such that if β<α then A(Mβ) embeds as a closed subgroup of A(Mα) but no subspace of A(Mβ) is homeomorphic to A(Mα).

We show that the character (minimal size of a local base at 0) of A(M) is d (minimal size of a cofinal set in NN) for every non-discrete, analytic M, but consistently there is a co-analytic M such that the character of A(M) is strictly above d.

The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an A(X), and a connection with the family of compact subsets of an auxiliary space.



中文翻译:

Tukey阶和自由阿贝尔拓扑群的多样性

对于吉洪诺夫空间X游离阿贝尔拓扑群超过X,表示为一种X,是集合X上具有最粗糙拓扑的自由Abelian群,因此对于X到Abelian拓扑群的任何连续映射,其正规扩展为一种X 是连续的。

我们证明有一个家庭 一种 最大尺寸 2C,由可分离的可量化空间组成,因此,如果MN一种 然后 一种中号一种ñ 不是拓扑同构的(此外, 一种中号 都没有拓扑嵌入 一种ñ 也不是 一种ñ)。我们显示有一个链C={中号αα<C+},最大大小,可分离的可量化空间,例如 β<α 然后 一种中号β 嵌入为的封闭子组 一种中号α 但没有子空间 一种中号β 是同胚的 一种中号α

我们证明了该字符(本地底数的最小大小为0) 一种中号d (在 ññ)对于每个非离散分析M,但始终存在一个协分析M,使得一种中号 严格以上 d

这些结果使用的主要工具是在 一种X,以及与辅助空间的紧凑子集族的连接。

更新日期:2021-02-18
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