  In this paper, we continue to study one of the classic problems in general
topology raised by P.S. Alexandrov: when a Hausdorff space $X$ has a continuous
bijection (a condensation) onto a compactum? We concentrate on the situation
when not only $X$ but also $X\setminus Y$ can be condensed onto a compactum
whenever the cardinality of $Y$ does not exceed certain $\tau$.
