Description
he concept of weakly separated spaces and the cardinal function $R$ was introduced by
Tkacenko, where the following question was also posed:
does $R(X)={\omega}$ (or even $R(X^{\omega})={\omega}$)
imply that $X$ has a countable network?
Several consistent counterexamples have been provided, but no ZFC counterexample is currently known.
The concept of $C({\omega}_1)$ was recently introduced by Alejandro Ríos-Herrejónin \,
providing further insight into the relationship between the assumptions
$R(X^{\omega})={\omega}$ and $\nw(X)={\omega}$.
We say that a topological space has property $C(\omega_1)$ iff
for any neighborhood assignment $U$ on a subset of $X$ with $|\dom(U)|=\omega_1 $,
there exists a set $Y\in {[\dom(U)]}^{{\omega_1}}$ such that
\begin{displaymath}
Y\subset \bigcap_{y\in Y}U(y).
\end{displaymath}
Clearly
$nw(X)={\omega}\ \to\ X\in C({\omega}_1) \to R(X^{\omega})={\omega}.$
We introduce additional notions that allow for a more detailed analysis of properties of certain
spaces obtained as counterexamples.