For a Hausdorff zero-dimensional topological space $X$ and a totally ordered field $F$ with interval topology, let $C_c(X,F)$ be the ring of all $F-$valued continuous functions on $X$ with countable range. It is proved that if $F$ is either an uncountable field or countable subfield of $\mathbb{R}$, then the structure space of $C_c(X,F)$ is $β_0X$, the Banaschewski Compactification of $X$. The ideals $\{O^{p,F}_c:p\in β_0X\}$ in $C_c(X,F)$ are introduced as modified countable analogue of the ideals $\{O^p:p\inβX\}$ in $C(X)$. It is realized that $C_c(X,F)\cap C_K(X,F)=\bigcap_{p\inβ_0X\texttt{\textbackslash}X} O^{p,F}_c$, this may be called a countable analogue of the well-known formula $C_K(X)=\bigcap_{p\inβX\texttt{\textbackslash}X}O^p$ in $C(X)$. Furthermore, it is shown that the hypothesis $C_c(X,F)$ is a Von-Neumann regular ring is equivalent to amongst others the condition that $X$ is a $P-$space.
In this paper we consider the question of when the space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology is countable dense homogeneous. In particular, we focus on the case when $X$ is countable with a unique non-isolated point $\infty$. In this case, $C_p(X)$ is countable dense homogeneous if and only if the filter of open neighborhoods of $\infty$ is a non-meager $P$-filter.
N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences（俄罗斯科学院乌拉尔分校 N.N.克拉索夫斯基数学力学研究所）；Department of General Topology and Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State University（莫斯科国立大学机械数学系一般拓扑与几何教研室）
A space $X$ is strongly $Y$-selective (resp., $Y$-selective) if every lower semicontinuous mapping from $Y$ to the nonempty subsets (resp., nonempty closed subsets) of $X$ has a continuous selection. We also call $X$ (strongly) $C$-selective if it is (strongly) $Y$-selective for any countable space $Y$, and (strongly) $L$-selective if it is (strongly) ($ω+1$)-selective. E. Michael showed that every first countable space is strongly $C$-selective. We extend this by showing that every $W$-space in the sense of the second author is strongly $C$-selective. We also show that every GO-space is $C$-s…
The aim of this paper is to study the Frechet-Urysohn property of the space $Q_p(X,\mathbb{R})$ of real-valued quasicontinuous functions, defined on a Hausdorff space $X$, endowed with the pointwise convergence topology. It is proved that under Suslin's Hypothesis, for an open Whyburn space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable. In particular, it is true in the class of first-countable regular spaces $X$. In ZFC, it is proved that for a metrizable space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable.
Sobriety of Regular Open Algebras in Second-Countable T3 Spaces
Yancheng Teachers University（盐城师范学院）；Beijing University of Posts and Telecommunications（北京邮电大学）；National Institute of Education, Nanyang Technological University（南洋理工大学国立教育学院）
A. V. Arhangel'skietheet在2012年，当他访问Abduaziz国王大学数学系时，介绍了新的较弱版本的常态，称为\it $C$ -常态，\rm和\it可数常态。本文的目的是研究可数正态性。我们证明了正规性蕴涵可数正规性，但反之一般不成立。我们提出了一些例子来显示可数正态性和其他较弱版本的正态性，如$C$ -normal，$L$ -noramlity，和轻度正态性之间的关系。我们回答以下公开问题的Arhangel'skietheet：“是否有一个吉洪诺夫空间，这不是PDMATHAMN-正常的？".在本文中，我们用$\langle x,y\rangle$表示有序对，用$\mathbb{N}$表示正整数集，用$\mathbb{R}$表示实数集。$T_4$空间是$T_1$正规空间，吉洪诺夫空间是$T_1$完全正则空间，而$T_3$空间是$T_1$正则空间。我们在紧性的定义中不假定$T_2$，在林德洛夫性的定义中不假定正则性。对于空间$X$的子集$A$，${\rm int} A$和$\overline{A}$分别表示$A$的内部和闭包。序数$γ$是所有序数$α$的集合，使得$α<γ$。第一个无穷序数是$ω_0$，第一个不可数序数是$ω_1$，$ω_1$的后继基数是$ω_2$。
A fundamental result proved by Bourgain, Fremlin and Talagrand states that the space $B_1(M)$ of Baire one functions over a Polish space $M$ is an angelic space. Stegall extended this result by showing that the class $B_1(M,E)$ of Baire one functions valued in a normed space $E$ is angelic. These results motivate our study of various topological properties in the classes $B_α(X,G)$ of Baire-$α$ functions, where $α$ is a nonzero countable ordinal, $G$ is a metrizable non-precompact abelian group and $X$ is a $G$-Tychonoff first countable space. In particular, we show that (1) $B_α(X,G)$ is a $κ…
Intermediate rings of real valued continuous functions with countable range on a Hausdorff zero-dimensional space $X$ are introduced in this article. Let $Σ_c(X)$ be the family of all such intermediate rings $A_c(X)$'s which lie between $C_c^*(X)$ and $C_c(X)$. It is shown that the structure space of each $A_c(X)$ is $β_0X$, the Banaschewski compactification of $X$. $X$ is shown to be a $P$-space if and only if each ideal in $C_c(X)$ is closed in the $m_c$-topology on it. Furthermore $X$ is realized to be an almost $P$-space when and only when each maximal ideal/ $z$-ideal in $C_c(X)$ becomes …
We give a short proof, that can be used in an introductory real analysis course, that if a function that is defined on the set of real numbers is continuous on a countable dense set, then it is continuous on an uncountable set. This is done for functions defined on complete metric spaces without isolated points, and the argument only uses that Cauchy sequences converge, and we prove the version related to Volterra's theorem. We discuss how this theorem is a direct consequence of the Baire category theorem, and also discuss Volterra's theorem and the history of this problem. We conclude with a …
Quotient Spaces Determined by Algebras of Continuous Functions
证明了若$X$是局部紧的$σ$ -紧空间，则在由$X$上所有实值有界连续函数的代数所确定的商$γ(X)$上，商拓扑与由该代数所定义的完全正则拓扑相等。由此推论，如果$X$是第二可数局部紧的，则$γ(X)$是第二可数局部紧的当且仅当它是第一可数的。对这些结果的兴趣起源于R. J. Archbold和S. Echterhoff和D. P. Williams在这里考虑了$C^*$ -代数的本原理想空间.
We prove that on the Baire space $(D^κ,π)$, $κ\geq ω_0$ where $D$ is a uniformly discrete space having $ω_1$-strongly compact cardinal and $π$ denotes the product uniformity on $D^κ$, there exists a $z_u$-filter $\mathcal{F}$ being Cauchy for the uniformity $eπ$ having as a base all the countable uniform partitions of $(D^κ,π)$, and failing the countable intersection property. This fact is equivalent to the existence of a non-vanishing real-valued uniformly continuous function $f$ on $D^κ$ for which the inverse function $g=1/f$ cannot be continuously extended to the completion of $(D^{κ_0},eπ)…
In this paper, two outwardly different graphs, namely, the zero divisor graph $Γ(C_c(X))$ and the comaximal graph $Γ_2^{'}(C_c(X))$ of the ring $C_c(X)$ of all real-valued continuous functions having countable range, defined on any Hausdorff zero dimensional space $X$, are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness. are concerned. However, the study reveals that the zero divisor graph $Γ(A_c(X))$ of an intermediate ring $A_c(X)$ of $C_c(X)$ is complemented if and only if the s…
