Sequences and dense sets

By the classical Hewitt–Marczewski–Pondiczery theorem (see [2], [3]) the Tychonoff product of 2ω many separable spaces is separable [2,3]. We consider the problem of the existence in the Tychonoff product of 2ω many separable spaces a dense countable subset Q, which contains no nontrivial convergent...

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Bibliographic Details
Published in:Topology and its applications 2020-02, Vol.271, p.106988, Article 106988
Main Author: Gryzlov, A.A.
Format: Article
Language:English
Subjects:
Convergent sequence
Dense set
Independent matrix
Projection
Tychonoff product
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Summary:By the classical Hewitt–Marczewski–Pondiczery theorem (see [2], [3]) the Tychonoff product of 2ω many separable spaces is separable [2,3]. We consider the problem of the existence in the Tychonoff product of 2ω many separable spaces a dense countable subset Q, which contains no nontrivial convergent sequences. We prove (Theorem 4.1) that such dense set exists in the product ∏α∈2ωZα of separable spaces {Zα:α∈2ω} if in every space Zα (α∈2ω) there are two closed disjoint not empty sets, we call such spaces decomposable. The class of decomposable spaces includes not single point T1-spaces, but also some T0-spaces and some spaces, which are not even T0-spaces.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2019.106988