Locally functionally countable subalgebra of ℛ ( L )

M. Elyasi; A. A. Estaji; M. Robat Sarpoushi

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 3, page 127-140
  • ISSN: 0044-8753

Abstract

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Let L c ( X ) = { f ∈ C ( X ) : C f ¯ = X } , where C f is the union of all open subsets U ⊆ X such that | f ( U ) | ≤ ℵ 0 . In this paper, we present a pointfree topology version of L c ( X ) , named ℛ ℓ c ( L ) . We observe that ℛ ℓ c ( L ) enjoys most of the important properties shared by ℛ ( L ) and ℛ c ( L ) , where ℛ c ( L ) is the pointfree version of all continuous functions of C ( X ) with countable image. The interrelation between ℛ ( L ) , ℛ ℓ c ( L ) , and ℛ c ( L ) is examined. We show that L c ( X ) ≅ ℛ ℓ c ( 𝔒 ( X ) ) for any space X . Frames L for which ℛ ℓ c ( L ) = ℛ ( L ) are characterized.

How to cite

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Elyasi, M., Estaji, A. A., and Robat Sarpoushi, M.. "Locally functionally countable subalgebra of $\mathcal {R}(L)$." Archivum Mathematicum 056.3 (2020): 127-140. <http://eudml.org/doc/297169>.

@article{Elyasi2020,
abstract = {Let $L_c(X)= \lbrace f \in C(X) \colon \overline\{C_f\}= X\rbrace $, where $C_f$ is the union of all open subsets $U \subseteq X$ such that $\vert f(U) \vert \le \aleph _0$. In this paper, we present a pointfree topology version of $L_c(X)$, named $\mathcal \{R\}_\{\ell c\}(L)$. We observe that $\mathcal \{R\}_\{\ell c\}(L)$ enjoys most of the important properties shared by $\mathcal \{R\}(L)$ and $\mathcal \{R\}_c(L)$, where $\mathcal \{R\}_c(L)$ is the pointfree version of all continuous functions of $C(X)$ with countable image. The interrelation between $\mathcal \{R\}(L)$, $\mathcal \{R\}_\{\ell c\}(L)$, and $\mathcal \{R\}_c(L)$ is examined. We show that $L_c(X) \cong \mathcal \{R\}_\{\ell c\}\big (\mathfrak \{O\}(X)\big )$ for any space $X$. Frames $L$ for which $\mathcal \{R\}_\{\ell c\}(L)=\mathcal \{R\}(L)$ are characterized.},
author = {Elyasi, M., Estaji, A. A., Robat Sarpoushi, M.},
journal = {Archivum Mathematicum},
keywords = {functionally countable subalgebra; locally functionally countable subalgebra; sublocale; frame},
language = {eng},
number = {3},
pages = {127-140},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Locally functionally countable subalgebra of $\mathcal \{R\}(L)$},
url = {http://eudml.org/doc/297169},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Elyasi, M.
AU - Estaji, A. A.
AU - Robat Sarpoushi, M.
TI - Locally functionally countable subalgebra of $\mathcal {R}(L)$
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 3
SP - 127
EP - 140
AB - Let $L_c(X)= \lbrace f \in C(X) \colon \overline{C_f}= X\rbrace $, where $C_f$ is the union of all open subsets $U \subseteq X$ such that $\vert f(U) \vert \le \aleph _0$. In this paper, we present a pointfree topology version of $L_c(X)$, named $\mathcal {R}_{\ell c}(L)$. We observe that $\mathcal {R}_{\ell c}(L)$ enjoys most of the important properties shared by $\mathcal {R}(L)$ and $\mathcal {R}_c(L)$, where $\mathcal {R}_c(L)$ is the pointfree version of all continuous functions of $C(X)$ with countable image. The interrelation between $\mathcal {R}(L)$, $\mathcal {R}_{\ell c}(L)$, and $\mathcal {R}_c(L)$ is examined. We show that $L_c(X) \cong \mathcal {R}_{\ell c}\big (\mathfrak {O}(X)\big )$ for any space $X$. Frames $L$ for which $\mathcal {R}_{\ell c}(L)=\mathcal {R}(L)$ are characterized.
LA - eng
KW - functionally countable subalgebra; locally functionally countable subalgebra; sublocale; frame
UR - http://eudml.org/doc/297169
ER -

References

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