Zero-dimensional mapping

A continuous mapping $f\colon X\to Y$ (where $X$ and $Y$ are topological spaces) such that $f^{-1}(y)$ is a zero-dimensional set (in the sense of $\operatorname{ind}$) for every $y\in Y$. 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 $X$ to $Y$ (or, more often, from $Y$ to $X$).

Example 1.

Every metric space $X$ with $\dim X\leq n$ admits a complete zero-dimensional mapping into a space $Y$ with a countable base and $\dim Y\leq n$ (Katetov's theorem). Here, complete zero-dimensionality means that for every $\epsilon>0$ and every $y\in f(X)$ there is a neighbourhood $U_y\subset Y$ whose inverse image $f^{-1}(U_y)$ splits into a discrete system of open sets in $X$ of diameter $<\epsilon$.

Example 2.

If a zero-dimensional mapping $f\colon X\to Y$, where $X$ is a normal locally connected space, is a perfect mapping, then the weight of $X$ is the same as that of $Y$ (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 $f\colon X\to Y$ is zero-dimensional, then $\dim Y\geq\dim X$. 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)