A Hausdorff space $(X, \tau)$ is said to be $k_w$-space if there is a countable collection $X_n$, $n \in \mathbb{N}$ of compact subsets of $X$, s.t.:
a) $X_n \subseteq X_{n+1}$, for all $n$
b) $X = \cup_{n \in \mathbb{N}}X_n$
c) any subset $A$ on $X$ is closed if and only if $A \cap X_n$ is compact for each $n \in \mathbb{N}$
Prove that every $k_w$ space is normal.
In order to show that space is normal - we need to show that for any closed disjoint subsets $A$ and $B$, there exist open sets $U$ and $V$, s.t. $A \subseteq U$ and $B \subseteq V$ and $U \cap V = \emptyset$. Let $A_i = A \cap X_i$, by property c) of $k_w$, $A_i$ is compact. Since $A_i$ and $B_i$ are disjoint compact sets, by Hausdorfness of $X$, there exist open $U_i$ and $V_i$, s.t. $A_i \subseteq U_i$ and $B_i \subseteq V_i$ and $U_i \cap V_i = \emptyset$.
But then how can I build $U$ and $V$? The problem is that $U_i$ can intersect $V_j$ for some $i \neq j$.
The answer is yes as I found in
Franklin, S.P. and B.V. Smith Thomas, A survey of k$_\omega$-spaces, Topology Proceedings 2 (1978), 111–124.
See https://pdfs.semanticscholar.org/8b2a/1b2db52abdf330df52fc019468002ffe093e.pdf p.113.