Relatively functionally countable subsets of products
| Autor: | Lipin, Anton |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A subset $A$ of a topological space $X$ is called relatively functionally countable (RFC) in $X$, if for each continuous function $f : X \to \mathbb{R}$ the set $f[A]$ is countable. We prove that all RFC subsets of a product $\prod\limits_{n\in\omega}X_n$ are countable, assuming that spaces $X_n$ are Tychonoff and all RFC subsets of every $X_n$ are countable. In particular, in a metrizable space every RFC subset is countable. The main tool in the proof is the following result: for every Tychonoff space $X$ and any countable set $Q \subseteq X$ there is a continuous function $f : X^\omega \to \mathbb{R}^2$ such that the restriction of $f$ to $Q^\omega$ is injective. Comment: 11 pages, minor changes |
| Databáze: | arXiv |
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