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Section
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Mathematics
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Title
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Products of spaces and the convergence of sequences
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Author(-s)
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Gryzlov A.A.a,
Golovastov R.A.a,
Bastrykov E.S.a
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Affiliations
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Udmurt State Universitya
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Abstract
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By the Hewitt–Marczewski–Pondiczery theorem, the Tychonoff product of $2^\omega$ separable spaces is separable. We continue to explore the problem of the existence in the Tychonoff product $\prod\limits_{\alpha\in 2^\omega}Z_\alpha$ of $2^\omega$ separable spaces a dense countable subset, which does not contain non-trivial convergent sequences. We say that a sequence $\lambda=\{x_n\colon n\in\omega\}$ is simple, if, for every $x_n\in\lambda$, a set $\{n'\in\omega\colon x_{n'}=x_n\}$ is finite. We prove that in the product of separable spaces $\prod\limits_{\alpha\in 2^\omega}Z_\alpha$, such that $Z_\alpha$ $(\alpha\in 2^\omega)$ contains a simple nonconvergent sequence, there is a countable dense set $Q\subseteq\prod\limits_{\alpha\in 2^\omega}Z_\alpha$, which does not contain non-trivial convergent in $\prod\limits_{\alpha\in 2^\omega}Z_\alpha$ sequences.
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Keywords
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Tychonoff product, dense set, convergent sequence, independent matrix
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UDC
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515.122
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MSC
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54A25, 54B10
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DOI
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10.35634/vm230402
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Received
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11 July 2023
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Language
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English
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Citation
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Gryzlov A.A., Golovastov R.A., Bastrykov E.S. Products of spaces and the convergence of sequences, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 4, pp. 563–570.
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