On Classes of Subcompact Spaces

This paper continues the study of P. S. Alexandroff's problem: When can a Hausdorff space $X$ be one-to-one continuously mapped onto a compact Hausdorff space? For a cardinal number $\tau$, the classes of $a_\tau$-spaces and strict $a_\tau$-spaces are defined. A compact space $X$ is called an $a_\tau$-space if, for any $C\in[X]^{\le\tau}$, there exists a one-to-one continuous mapping of $X\setminus C$ onto a compact space. A compact space $X$ is called a strict $a_\tau$-space if, for any $C\in[X]^{\le\tau}$, there exits a one-to-one continuous mapping of $X\setminus C$ onto a compact space $Y$, and this mapping can be continuously extended to the whole space $X$. In this paper, we study properties of the classes of $a_\tau$- and strict $a_\tau$-spaces by using Raukhvarger's method of special continuous paritions.