Unboring ideals

We say that a space $X$ is $FinBW(I)$ ($I$ is an ideal on the set of natural numbers), if for each sequence $(x_n)$ in $X$ one can find a set $A$ not belonging to $I$ such that $(x_n)_{n\in A}$ converges in $X$. Thus, the classical Bolzano-Weierstrass theorem states that every compact subset of the real line is $FinBW(Fin)$ ($Fin$ is the ideal of all finite subsets of naturals). During my talk I will present new results concerning $FinBW(I)$ spaces and discuss relationship between the studied notions and the Katetov order on ideals. In particular, under $MA$ I will characterize for all $Pi^0_4$ ideals when $FinBW(I)$ and $FinBW(J)$ differ.