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- W4299827892 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 $C_c^{K}(X)=O_c^{beta_0Xsetminus X}$ (resp., $C_c^{psi}(X)=M_c^{beta_0Xsetminus upsilon_0X}$). This implies that for an $Bbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ equals to $C_c^K(X)$, i.e., $M_c^{beta_0Xsetminus X}=C_c^K(X)$. Afterwards, by applying methods of functionally countable subalgebras, we observe some results in the remainder of Banaschewski compactification. It is shown that for a zero-dimensional non pseudocompact space $X$, the set $beta_0Xsetminus upsilon_0X$ has cardinality at least $2^{2^{aleph_0}}$. Moreover, for a locally compact and $Bbb{N}$-compact space $X$, the remainder $beta_0Xsetminus X$ is an almost $P$-space. These results leads us to find a class of Parovi$breve{mbox{c}}$enko spaces in Banaschewski compactification os a non pseudocompact zero-dimensional space. We conclude with a theorem which gives a lower bound for the cellularity of subspaces $beta_0Xsetminus upsilon_0X$ and $beta_0Xsetminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact." @default.
- W4299827892 created "2022-10-03" @default.
- W4299827892 creator A5075396672 @default.
- W4299827892 date "2015-06-30" @default.
- W4299827892 modified "2023-09-25" @default.
- W4299827892 title "Functionally countable subalgebras and some properties of Banaschewski compactification" @default.
- W4299827892 doi "https://doi.org/10.48550/arxiv.1506.08980" @default.
- W4299827892 hasPublicationYear "2015" @default.
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