  We investigate Noetherian families. By using a special Noetherian $\pi$-base,
we give a result which states that existence of NONEMPTY's 2-tactic in the
Banach-Mazur game on a space $X$, BM(X), if NONEMPTY has a winning strategy in
$BM(X)$ and $X$ has the special Noetherian $\pi$-base. This result includes one
of the Galvin's theorems which is important in this topic. From this result, we
prove that for any topological space $X$ if $\pi w(X)\leq \omega_1$ and
NONEMPTY has a winning strategy in $BM(X)$, then NONEMPTY has a 2-tactic in
$BM(X)$. As a result of this fact, under $\mathsf{CH}$, we show that for any
separable $T_3$ space $X$ if NONEMPTY has a winning strategy in $BM(X)$, then
NONEMPTY has a 2-tactic in $BM(X)$. So, we prove that Telg{\'a}rsky's
conjecture cannot be proven true in the realm of separable $T_3$ spaces, and
more generally, in the class of spaces $X$ with $\pi w(X)\leq \omega_1$. We
pose some questions about this topic.
