Perfect mapping

A continuous closed mapping (cf. Closed mapping; Continuous mapping) of topological spaces under which the pre-image of every point is compact. Perfect mappings are akin to continuous mappings from compact spaces into Hausdorff spaces (every such mapping is perfect), although the scope of the definition covers all topological spaces. In the class of completely-regular spaces, the perfect mappings are characterized by the fact that their Stone–Čech extension maps remainders to remainders (cf. Remainder of a space). Perfect mappings preserve metrizability, paracompactness, weight, and Čech completeness. Other invariants (such as the character of a space) are transformed in a proper way. The class of perfect mappings is closed under taking products and composition. A restriction of a perfect mapping to a closed subspace is perfect (this is false for quotient and open mappings).

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.

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[2]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[3]	R. Engelking, "General topology" , Heldermann (1989)