Some Properties of Subcompact Spaces

A Hausdorff topological space $$X$$ is said to be subcompact if it admits a coarser compact Hausdorff topology. P. S. Alexandroff asked the following question: What Hausdorff spaces are subcompact? A compact space $$X$$ is called a strict $$a$$ -space if, for any $$C\in [X]^{\le\omega}$$ , there exists a one-to-one continuous map of $$X\setminus C$$ onto a compact space $$Y$$ which can be continuously extended to the entire space $$X$$ . The paper continues the study of classes of subcompact spaces. It is proved that the product of a compact space and a dyadic compact space without isolated points is a strict $$a$$ -space.