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