View source for Zero-dimensional mapping

A [[Continuous mapping|continuous mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992101.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992103.png" /> are topological spaces) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992104.png" /> is a zero-dimensional set (in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992105.png" />) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992106.png" />. 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|Cardinal characteristic]]) transfer from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992107.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992108.png" /> (or, more often, from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z0992109.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921010.png" />).

===Example 1.===
Every metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921012.png" /> admits a complete zero-dimensional mapping into a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921013.png" /> with a countable base and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921014.png" /> (Katetov's theorem). Here, complete zero-dimensionality means that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921015.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921016.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921017.png" /> whose inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921018.png" /> splits into a discrete system of open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921019.png" /> of diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921020.png" />.

===Example 2.===
If a zero-dimensional mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921022.png" /> is a normal locally connected space, is a [[Perfect mapping|perfect mapping]], then the weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921023.png" /> is the same as that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921024.png" /> (cf. [[Weight of a topological space|Weight of a topological space]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   B.A. Pasynkov,   "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR></table>



====Comments====
The starting point for studying zero-dimensional mappings was the theorem in compact metric spaces that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921025.png" /> is zero-dimensional, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099210/z09921026.png" />. It extends to separable metric spaces for closed continuous mappings, but not for open ones; see [[#References|[a1]]], p. 91.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hurevicz,   G. Wallman,   "Dimension theory" , Princeton Univ. Press  (1948)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR></table>