  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
$\omega_2$.
