| Title:
|
Functionally countable subalgebras and some properties of the Banaschewski compactification (English) |
| Author:
|
Olfati, A. R. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
57 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
365-379 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
zero-dimensional space |
| Keyword:
|
strongly zero-dimensional space |
| Keyword:
|
$\mathbb{N}$-compact space |
| Keyword:
|
Banaschewski compactification |
| Keyword:
|
pseudocompact space |
| Keyword:
|
functionally countable subalgebra |
| Keyword:
|
support |
| Keyword:
|
cellularity |
| Keyword:
|
remainder |
| Keyword:
|
almost $P$-space |
| Keyword:
|
Parovičenko space |
| MSC:
|
54A25 |
| MSC:
|
54C30 |
| MSC:
|
54C40 |
| MSC:
|
54D40 |
| MSC:
|
54D60 |
| MSC:
|
54G05 |
| idZBL:
|
Zbl 1374.54030 |
| idMR:
|
MR3554517 |
| DOI:
|
10.14712/1213-7243.2015.170 |
| . |
| Date available:
|
2016-09-22T15:29:20Z |
| Last updated:
|
2018-10-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145841 |
| . |
| Reference:
|
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| . |
