Zero-dimensional mapping

A continuous mapping (where and are topological spaces) such that is a zero-dimensional set (in the sense of ) for every . The application of zero-dimensional and closely related mappings reduces the study of a given space to that of another, simpler, one. Thus, many dimension properties and other cardinal invariants (cf. Cardinal characteristic) transfer from to (or, more often, from to ).

Example 1.

Every metric space with admits a complete zero-dimensional mapping into a space with a countable base and (Katetov's theorem). Here, complete zero-dimensionality means that for every and every there is a neighbourhood whose inverse image splits into a discrete system of open sets in of diameter .

Example 2.

If a zero-dimensional mapping , where is a normal locally connected space, is a perfect mapping, then the weight of is the same as that of (cf. Weight of a topological space).

References

[1]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)

Comments

The starting point for studying zero-dimensional mappings was the theorem in compact metric spaces that if is zero-dimensional, then . It extends to separable metric spaces for closed continuous mappings, but not for open ones; see [a1], p. 91.

References

[a1]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
[a2]	R. Engelking, "General topology" , Heldermann (1989)