Every scattered space is subcompact

We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every Čech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω-monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense Gδ-subsets of Cantor cubes are subcompact.