Compact condensations of Hausdorff spaces

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\Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\setminus Y$$\end{document} can be condensed onto a compactum whenever the cardinality of Y does not exceed certain τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document}.