Functionally countable subalgebras and some properties of Banaschewski compactification

TL;DR

This paper explores properties of functionally countable subalgebras and the Banaschewski compactification of zero-dimensional spaces, revealing cardinality bounds and structural characteristics of the remainders.

Contribution

It introduces new relationships between functionally countable subalgebras and Banaschewski compactification, including cardinality and topological properties of the remainders.

Findings

The intersection of all free maximal ideals in $C_c(X)$ equals $C_c^K(X)$ for $N$-compact spaces.

The remainder $eta_0Xackslash u_0X$ has cardinality at least $2^{2^{ ext{aleph}_0}}$ for non pseudocompact spaces.

The remainder $eta_0Xackslash X$ is an almost $P$-space for locally compact, $N$-compact spaces.

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., $C_c^{\psi}(X)=M_c^{\beta_0X\setminus \upsilon_0X}$). 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., $M_c^{\beta_0X\setminus X}=C_c^K(X)$. 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 $\beta_0X\setminus \upsilon_0X$ has cardinality at least $2^{2^{\aleph_0}}$. Moreover, for a locally compact and $\Bbb{N}$-compact space $X$, the remainder $\beta_0X\setminus X$ 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 $\beta_0X\setminus \upsilon_0X$ and $\beta_0X\setminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact.