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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2406.03014 |
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| _version_ | 1866917722581893120 |
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| author | Osipov, Alexander V. |
| author_facet | Osipov, Alexander V. |
| contents | A space $X$ is sequentially separable if there is a countable $S\subset X$ such that every point of $X$ is the limit of a sequence of points from $S$. In 2004, N.V. Velichko defined and investigated concepts close to sequentially separability: $σ$-separability and $F$-separability. The aim of this paper is to study $σ$-separability and $F$-separability (and their hereditary variants) of the space $C_p(X)$ of all real-valued continuous functions, defined on a Tychonoff space $X$, endowed with the pointwise convergence topology. In particular, we proved that $σ$-separability coincides with sequential separability. Hereditary variants (hereditarily $σ$-separablity and hereditarily $F$-separablity) coincides with Frechet-Urysohn property in the class of cosmic spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03014 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Velichko's notions close to sequentially separability and their hereditary variants in $C_p$-theory Osipov, Alexander V. General Topology A space $X$ is sequentially separable if there is a countable $S\subset X$ such that every point of $X$ is the limit of a sequence of points from $S$. In 2004, N.V. Velichko defined and investigated concepts close to sequentially separability: $σ$-separability and $F$-separability. The aim of this paper is to study $σ$-separability and $F$-separability (and their hereditary variants) of the space $C_p(X)$ of all real-valued continuous functions, defined on a Tychonoff space $X$, endowed with the pointwise convergence topology. In particular, we proved that $σ$-separability coincides with sequential separability. Hereditary variants (hereditarily $σ$-separablity and hereditarily $F$-separablity) coincides with Frechet-Urysohn property in the class of cosmic spaces. |
| title | Velichko's notions close to sequentially separability and their hereditary variants in $C_p$-theory |
| topic | General Topology |
| url | https://arxiv.org/abs/2406.03014 |