Countable Normality

Abstract

A. V. Arhangel'ski\u{i} introduced in 2012, when he was visiting the department of Mathematics at King Abduaziz University, new weaker versions of normality, called \it $C$-normality, \rm and \it countable normality. \rm The purpose of this paper is to investigate countable normality property. We prove that normality implies countable normality but the converse is not true in general. We present some examples to show relationships between countable normality and other weaker versions of normality such as $C$-normality, $L$-noramlity, and mild normality. We answer the following open problem of Arhangel'ski\u{i}: "Is there a Tychonoff space which is not $C$-normal ?". Throughout this paper, we denote an ordered pair by $\langle x,y\rangle$, the set of positive integers by $\mathbb{N}$ and the set of real numbers by $\mathbb{R}$. A $T_4$ space is a $T_1$ normal space, a Tychonoff space is a $T_1$ completely regular space, and a $T_3$ space is a $T_1$ regular space. We do not assume $T_2$ in the definition of compactness and we do not assume regularity in the definition of Lindel\"ofness. For a subset $A$ of a space $X$, ${\rm int} A$ and $\overline{A}$ denote the interior and the closure of $A$, respectively. An ordinal $\gamma$ is the set of all ordinals $\alpha$ such that $\alpha<\gamma$. The first infinite ordinal is $\omega_0$, the first uncountable ordinal is $\omega_1$, and the successor cardinal of $\omega_1$ is $\omega_2$.