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''large inductive dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507601.png" />, small inductive dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507602.png" />''
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{{TEX|done}}{{MSC|54F45}}
  
Dimension invariants (cf. [[Dimension invariant|Dimension invariant]]) of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507603.png" />; both are defined by means of the notion of a partition between two sets. The definition is by induction, as follows. For the empty space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507604.png" /> one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507605.png" />. Under the hypothesis that all spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507606.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507607.png" /> are known, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507608.png" /> is a non-negative integer, one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i0507609.png" /> if for any two disjoint closed subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076012.png" /> there is a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076013.png" /> between them for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076014.png" />. Here, a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076015.png" /> is called a [[Partition|partition]] between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076018.png" /> if the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076019.png" /> is the sum of two open disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076021.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076023.png" />, respectively. This definition transfers to the definition of small inductive dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076024.png" /> by taking one of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076025.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076026.png" /> to consist of a single point, while the other is an arbitrary closed set not containing this point.
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''large inductive dimension $\mathrm{Ind}\,X$, small inductive dimension $\mathrm{ind}\,X$''
  
The large inductive dimension was defined for a fairly wide class of (metric) spaces by L.E.J. Brouwer [[#References|[1]]]. The small inductive dimension was defined independently by P.S. Urysohn [[#References|[2]]] and K. Menger [[#References|[3]]]. The study of inductive dimensions and, more generally, of dimension invariants, is only of interest under the hypothesis that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050760/i05076027.png" /> satisfies sufficiently strong separation axioms, mainly the axiom of normality (cf. [[Separation axiom|Separation axiom]]).
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[[Dimension invariant]]s of a topological space $X$; both are defined by means of the notion of a partition between two sets. The definition is by induction, as follows. For the empty space $X = \emptyset$ one sets $\mathrm{Ind}\,\emptyset = \mathrm{ind}\,\emptyset = -1$. Under the hypothesis that all spaces $X$ for which $\mathrm{Ind}\,X < n$ are known, where $n$ is a non-negative integer, one puts $\mathrm{Ind}\,X < n+1$ if for any two disjoint closed subsets $A$ and $B$ of $X$ there is a partition $C$ between them for which $\mathrm{Ind}\,C < n$. Here, a closed set $C$ is called a [[partition]] between $A$ and $B$ in $X$ if the open set $X \setminus C$ is the sum of two open disjoint sets $U_A$ and $U_B$ containing $A$ and $B$, respectively. This definition transfers to the definition of small inductive dimension $\mathrm{ind}\,X$ by taking one of the sets $A$ or $B$ to consist of a single point, while the other is an arbitrary closed set not containing this point.
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The large inductive dimension was defined for a fairly wide class of (metric) spaces by L.E.J. Brouwer [[#References|[1]]]. The small inductive dimension was defined independently by P.S. Urysohn [[#References|[2]]] and K. Menger [[#References|[3]]]. The study of inductive dimensions and, more generally, of dimension invariants, is only of interest under the hypothesis that the space $X$ satisfies sufficiently strong [[separation axiom]]s, mainly the axiom of [[Normal space|normality]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,  "Ueber den natürlichen Dimensionsbegriff"  ''J. Reine Angew. Math.'' , '''142'''  (1913)  pp. 146–152</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Urysohn,  "Les multiplicités cantoriennes"  ''C.R. Acad. Sci.'' , '''175'''  (1922)  pp. 440–442</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Menger,  "Ueber die Dimensionalität von Punktmengen. I"  ''Monatshefte Math. und Phys.'' , '''33'''  (1923)  pp. 148–160</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,  "Ueber den natürlichen Dimensionsbegriff"  ''J. Reine Angew. Math.'' , '''142'''  (1913)  pp. 146–152</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Urysohn,  "Les multiplicités cantoriennes"  ''C.R. Acad. Sci.'' , '''175'''  (1922)  pp. 440–442</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  K. Menger,  "Ueber die Dimensionalität von Punktmengen. I"  ''Monatshefte Math. und Phys.'' , '''33'''  (1923)  pp. 148–160</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. van Mill,  "Infinite-dimensional topology, prerequisites and introduction" , North-Holland  (1988)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. van Mill,  "Infinite-dimensional topology, prerequisites and introduction" , North-Holland  (1988)</TD></TR>
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</table>

Latest revision as of 19:38, 14 December 2016

2020 Mathematics Subject Classification: Primary: 54F45 [MSN][ZBL]

large inductive dimension $\mathrm{Ind}\,X$, small inductive dimension $\mathrm{ind}\,X$

Dimension invariants of a topological space $X$; both are defined by means of the notion of a partition between two sets. The definition is by induction, as follows. For the empty space $X = \emptyset$ one sets $\mathrm{Ind}\,\emptyset = \mathrm{ind}\,\emptyset = -1$. Under the hypothesis that all spaces $X$ for which $\mathrm{Ind}\,X < n$ are known, where $n$ is a non-negative integer, one puts $\mathrm{Ind}\,X < n+1$ if for any two disjoint closed subsets $A$ and $B$ of $X$ there is a partition $C$ between them for which $\mathrm{Ind}\,C < n$. Here, a closed set $C$ is called a partition between $A$ and $B$ in $X$ if the open set $X \setminus C$ is the sum of two open disjoint sets $U_A$ and $U_B$ containing $A$ and $B$, respectively. This definition transfers to the definition of small inductive dimension $\mathrm{ind}\,X$ by taking one of the sets $A$ or $B$ to consist of a single point, while the other is an arbitrary closed set not containing this point.

The large inductive dimension was defined for a fairly wide class of (metric) spaces by L.E.J. Brouwer [1]. The small inductive dimension was defined independently by P.S. Urysohn [2] and K. Menger [3]. The study of inductive dimensions and, more generally, of dimension invariants, is only of interest under the hypothesis that the space $X$ satisfies sufficiently strong separation axioms, mainly the axiom of normality.

References

[1] L.E.J. Brouwer, "Ueber den natürlichen Dimensionsbegriff" J. Reine Angew. Math. , 142 (1913) pp. 146–152
[2] P.S. Urysohn, "Les multiplicités cantoriennes" C.R. Acad. Sci. , 175 (1922) pp. 440–442
[3] K. Menger, "Ueber die Dimensionalität von Punktmengen. I" Monatshefte Math. und Phys. , 33 (1923) pp. 148–160
[4] P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)


Comments

An extensive treatment of the subject can be found in [a1]. For a quick introduction to the dimension theory of separable metric spaces, see [a2], Chapt. 4.

References

[a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978)
[a2] J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988)
How to Cite This Entry:
Inductive dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inductive_dimension&oldid=13316
This article was adapted from an original article by V.I. Zaitsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article