# Perfect mapping

The above properties of perfect mappings have led to a situation where this class of mappings has begun to play a pivotal role in the classification of topological spaces. The completely-regular pre-images of metric spaces under perfect mappings are characterized as paracompact feathered $p$-) spaces (cf. Paracompact space; Feathered space). The class of paracompact $p$-spaces is closed under perfect mappings and their inverses. An important property of perfect mappings is that they can be restricted to certain closed subspaces without reducing the image in such a way that the resulting mapping is irreducible, that is, it cannot be further restricted without reducing the image (cf. also Irreducible mapping). Irreducible perfect mappings are the starting point for constructing a theory of absolutes of topological spaces (cf. Absolute). For an irreducible perfect mapping, the $\pi$-weight (cf. Weight of a topological space) of the image is always equal to that of the pre-image, and the Suslin number of the image is equal to that of the pre-image. If a completely-regular $T_1$-space is mapped onto a completely-regular $T_1$-space by a perfect mapping, then $X$ is homeomorphic to a closed subspace of the topological product of $Y$ with some $T_2$-compactum. The diagonal product of a perfect mapping and a continuous mapping of $T_2$-spaces is always a perfect mapping; in particular, the diagonal product of a perfect mapping and a compression (i.e. a one-to-one continuous mapping onto) is a homeomorphism. If a topological space can be mapped perfectly onto one metric space and compressed onto another metric space (which need not be the same), then it is itself metrizable.