Products of spaces and the convergence of sequences

A. A. Gryzlov ; R. A. Golovastov ; E. S. Bastrykov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 4, pp. 563-570

Résumé

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.

Détail

Keywords: Tychonoff product, dense set, convergent sequence, independent matrix