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.