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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.17892 |
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| _version_ | 1866910889748201472 |
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| author | Pol, Elżbieta Pol, Roman Reńska, Mirosława |
| author_facet | Pol, Elżbieta Pol, Roman Reńska, Mirosława |
| contents | We prove that if a separable metrizable $X$ is a union of two disjoint 0-dimensional sets $E$, $F$, $E$ is absolutely $G_δ$ and $F$ is absolutely $F_{σδ}$ then there is a closed embedding $h$ into the union of countable products of the irrationals and the rationals with $E$ being the preimage under $h$ of the countable product of the irrationals and $F$ being the preimage under $h$ of the countable product of the rationals. We prove also that for the set $H$ of points $x$ in the Hilbert cube such that for each $k$ there is $l$ with $x(2^k 3^l)=0$, whenever $A$ is an $F_{σδ}$ set in a compact one-dimensional space $X$, there is an embedding $h$ into the union of the countable product of the irrationals with added point $0$, and the countable product of the rationals, such that $A$ is the preimage under $h$ of the set $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On closed embeddings in $P^N \cup Q^N$ Pol, Elżbieta Pol, Roman Reńska, Mirosława General Topology We prove that if a separable metrizable $X$ is a union of two disjoint 0-dimensional sets $E$, $F$, $E$ is absolutely $G_δ$ and $F$ is absolutely $F_{σδ}$ then there is a closed embedding $h$ into the union of countable products of the irrationals and the rationals with $E$ being the preimage under $h$ of the countable product of the irrationals and $F$ being the preimage under $h$ of the countable product of the rationals. We prove also that for the set $H$ of points $x$ in the Hilbert cube such that for each $k$ there is $l$ with $x(2^k 3^l)=0$, whenever $A$ is an $F_{σδ}$ set in a compact one-dimensional space $X$, there is an embedding $h$ into the union of the countable product of the irrationals with added point $0$, and the countable product of the rationals, such that $A$ is the preimage under $h$ of the set $H$. |
| title | On closed embeddings in $P^N \cup Q^N$ |
| topic | General Topology |
| url | https://arxiv.org/abs/2503.17892 |