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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" />).
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A [[Continuous mapping|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|Cardinal characteristic]]) transfer from $X$ to $Y$ (or, more often, from $Y$ to $X$).
  
 
===Example 1.===
 
===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" />.
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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.===
 
===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]]).
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If a zero-dimensional mapping $f\colon X\to Y$, where $X$ is a normal locally connected space, is a [[Perfect mapping|perfect mapping]], then the weight of $X$ is the same as that of $Y$ (cf. [[Weight of a topological space|Weight of a topological space]]).
  
 
====References====
 
====References====
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====Comments====
 
====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.
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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 [[#References|[a1]]], p. 91.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:08, 21 November 2018

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)
How to Cite This Entry:
Zero-dimensional mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zero-dimensional_mapping&oldid=15288
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article