  Our main object of interest is the following notion: we say that a
topological space space $X$ is in FinBW($\mathcal{I}$), where $\mathcal{I}$ is
an ideal on $\omega$, if for each sequence $(x_n)_{n\in\omega}$ in $X$ one can
find an $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ converges in $X$.
  We define an ideal $\mathcal{BI}$ which is critical for FinBW($\mathcal{I}$)
in the following sense: Under CH, for every ideal $\mathcal{I}$,
$\mathcal{BI}\not\leq_K\mathcal{I}$ ($\leq_K$ denotes the Kat\v{e}tov preorder
of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$).
We show that $\mathcal{BI}\not\leq_K\mathcal{I}$ and $\omega_1$ with the order
topology is in FinBW($\mathcal{I}$), for all $\bf{\Pi^0_4}$ ideals
$\mathcal{I}$.
  We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is
nonempty: we prove under MA($\sigma$-centered) that for $\bf{\Pi^0_4}$ ideals
$\mathcal{I}$ and $\mathcal{J}$ this is equivalent to
$\mathcal{J}\not\leq_K\mathcal{I}$. Moreover, answering in negative a question
of M. Hru\v{s}\'ak and D. Meza-Alc\'antara, we show that the ideal
$\text{Fin}\times\text{Fin}$ is not critical among Borel ideals for
extendability to a $\bf{\Pi^0_3}$ ideal. Finally, we apply our results in
studies of Hindman spaces and in the context of analytic P-ideals.
