Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.19448 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916543591350272 |
|---|---|
| author | Estaji, Ali Akbar Taha, Maryam |
| author_facet | Estaji, Ali Akbar Taha, Maryam |
| contents | Let $\mathcal C_{c}(L):= \{α\in \mathcal{R}(L) \mid R_α \, \text{ is a countable subset of } \, \mathbb R \}$, where $R_α:=\{r\in\mathbb R \mid {\mathrm{coz}}(α-r)\neq\top\}$ for every $α\in\mathcal R (L).$ By using idempotent elements, it is going to prove that ${\mathrm{Coz}}_c[L]:= \{\mathrm{coz}(α) \mid α\in\mathcal{C}_c (L) \}$ is a $σ$-frame for every completely regular frame $L,$ and from this, we conclude that it is regular, paracompact, perfectly normal and an Alexandroff algebra frame such that each cover of it is shrinkable. Also, we show that $L$ is a zero-dimensional frame if and only if $ L$ is a $c$-completely regular frame. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Cozero part of the pointfree version of $C_c (X)$ Estaji, Ali Akbar Taha, Maryam General Topology Rings and Algebras Primary: 06D22, Secondary: 54C05, 54C30, 17C27 Let $\mathcal C_{c}(L):= \{α\in \mathcal{R}(L) \mid R_α \, \text{ is a countable subset of } \, \mathbb R \}$, where $R_α:=\{r\in\mathbb R \mid {\mathrm{coz}}(α-r)\neq\top\}$ for every $α\in\mathcal R (L).$ By using idempotent elements, it is going to prove that ${\mathrm{Coz}}_c[L]:= \{\mathrm{coz}(α) \mid α\in\mathcal{C}_c (L) \}$ is a $σ$-frame for every completely regular frame $L,$ and from this, we conclude that it is regular, paracompact, perfectly normal and an Alexandroff algebra frame such that each cover of it is shrinkable. Also, we show that $L$ is a zero-dimensional frame if and only if $ L$ is a $c$-completely regular frame. |
| title | The Cozero part of the pointfree version of $C_c (X)$ |
| topic | General Topology Rings and Algebras Primary: 06D22, Secondary: 54C05, 54C30, 17C27 |
| url | https://arxiv.org/abs/2412.19448 |