Functionally countable subalgebras and some properties of the Banaschewski compactification

@article{Olfati2016,
abstract = {Let $X$ be a zero-dimensional space and $C_c(X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K(X)$ (resp., $C_c^\{\psi \}(X))$ the set of all functions in $C_c(X)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K(X)=O_c^\{\beta _0X\setminus X\}$ (resp., $C^\{\psi \}_c(X)=M_c^\{\beta _0X\setminus \upsilon _0X\}$), where $\beta _0X$ is the Banaschewski compactification of $X$ and $\upsilon _0X$ is the $\mathbb \{N\}$-compactification of $X$. This implies that for an $\mathbb \{N\}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ is equal to $C_c^K(X)$, i.e., $M_c^\{\beta _0X\setminus X\}=C_c^K(X)$. By applying methods of functionally countable subalgebras, we then obtain some results in the remainder of the Banaschewski compactification. We show that for a non-pseudocompact zero-dimensional space $X$, the set $\beta _0X\setminus \upsilon _0X$ has cardinality at least $2^\{2^\{\aleph _0\}\}$. Moreover, for a locally compact and $\mathbb \{N\}$-compact space $X$, the remainder $\beta _0X\setminus X$ is an almost $P$-space. These results lead us to find a class of Parovičenko spaces in the Banaschewski compactification of a non pseudocompact zero-dimensional space. We conclude with a theorem which gives a lower bound for the cellularity of the subspaces $\beta _0X\setminus \upsilon _0X$ and $\beta _0X\setminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact.},
author = {Olfati, A. R.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {zero-dimensional space; strongly zero-dimensional space; $\mathbb \{N\}$-compact space; Banaschewski compactification; pseudocompact space; functionally countable subalgebra; support; cellularity; remainder; almost $P$-space; Parovičenko space},
language = {eng},
number = {3},
pages = {365-379},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Functionally countable subalgebras and some properties of the Banaschewski compactification},
url = {http://eudml.org/doc/286833},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Olfati, A. R.
TI - Functionally countable subalgebras and some properties of the Banaschewski compactification
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 3
SP - 365
EP - 379
AB - Let $X$ be a zero-dimensional space and $C_c(X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K(X)$ (resp., $C_c^{\psi }(X))$ the set of all functions in $C_c(X)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K(X)=O_c^{\beta _0X\setminus X}$ (resp., $C^{\psi }_c(X)=M_c^{\beta _0X\setminus \upsilon _0X}$), where $\beta _0X$ is the Banaschewski compactification of $X$ and $\upsilon _0X$ is the $\mathbb {N}$-compactification of $X$. This implies that for an $\mathbb {N}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ is equal to $C_c^K(X)$, i.e., $M_c^{\beta _0X\setminus X}=C_c^K(X)$. By applying methods of functionally countable subalgebras, we then obtain some results in the remainder of the Banaschewski compactification. We show that for a non-pseudocompact zero-dimensional space $X$, the set $\beta _0X\setminus \upsilon _0X$ has cardinality at least $2^{2^{\aleph _0}}$. Moreover, for a locally compact and $\mathbb {N}$-compact space $X$, the remainder $\beta _0X\setminus X$ is an almost $P$-space. These results lead us to find a class of Parovičenko spaces in the Banaschewski compactification of a non pseudocompact zero-dimensional space. We conclude with a theorem which gives a lower bound for the cellularity of the subspaces $\beta _0X\setminus \upsilon _0X$ and $\beta _0X\setminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact.
LA - eng
KW - zero-dimensional space; strongly zero-dimensional space; $\mathbb {N}$-compact space; Banaschewski compactification; pseudocompact space; functionally countable subalgebra; support; cellularity; remainder; almost $P$-space; Parovičenko space
UR - http://eudml.org/doc/286833
ER -