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Difference between revisions of "Local dimension"

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''of a normal topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601201.png" />''
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''of a normal topological space $X$''
  
The topological invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601202.png" />, defined as follows. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601204.png" /> if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601205.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601206.png" /> for which the [[Lebesgue dimension|Lebesgue dimension]] of its closure satisfies the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601207.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601208.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601209.png" />, then the local dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012010.png" /> is finite, so one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012011.png" /> and puts
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The topological invariant $\mathrm{locdim}(X)$, defined as follows. One says that $\mathrm{locdim}(X) \le n$, $n = -1,0,1,\ldots$ if for any point $x \in X$ there is a neighbourhood $O_x$ for which the [[Lebesgue dimension|Lebesgue dimension]] of its closure satisfies the relation $\dim \bar O_x \le n$. If $\mathrm{locdim}(X) \le n$ for some $n$, then the local dimension of $X$ is finite, so one writes $\mathrm{locdim}(X) < \infty$ and puts
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$$
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\mathrm{locdim}(X) = \min\{ n : \mathrm{locdim}(X) \le n \}
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012012.png" /></td> </tr></table>
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Always $\mathrm{locdim}(X) \le \dim(X)$; there are normal spaces $X$ with $\mathrm{locdim}(X) < \dim(X)$; in the class of paracompact spaces always $\mathrm{locdim}(X) = \dim(X)$. If in the definition of local dimension the Lebesgue dimension $\dim \bar O_x $ is replaced by the large [[Inductive dimension|inductive dimension]] $\mathrm{Ind} \bar O_x $, then one obtains the definition of the local large inductive dimension $\mathrm{locInd}(X)$.
 
 
Always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012013.png" />; there are normal spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012015.png" />; in the class of paracompact spaces always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012016.png" />. If in the definition of local dimension the Lebesgue dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012017.png" /> is replaced by the large [[Inductive dimension|inductive dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012018.png" />, then one obtains the definition of the local large inductive dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012019.png" />.
 
  
 
====Comments====
 
====Comments====
See [[#References|[a1]]] for a construction of a space with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012020.png" /> and — as an application — a hereditarily normal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012021.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012022.png" /> yet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l06012023.png" /> contains subspaces of arbitrary high dimension.
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See [[#References|[a1]]] for a construction of a space with $\mathrm{locdim}(X) < \dim(X)$ and — as an application — a hereditarily normal space $Y$ with $\dim Y = 0$ yet $Y$ contains subspaces of arbitrary high dimension.
  
 
For the notions of the local dimension at a point of an analytic space, algebraic variety or scheme cf. [[Analytic space|Analytic space]]; [[Dimension|Dimension]] of an associative ring; [[Analytic set|Analytic set]], and [[Spectrum of a ring|Spectrum of a ring]].
 
For the notions of the local dimension at a point of an analytic space, algebraic variety or scheme cf. [[Analytic space|Analytic space]]; [[Dimension|Dimension]] of an associative ring; [[Analytic set|Analytic set]], and [[Spectrum of a ring|Spectrum of a ring]].
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)  pp. 19; 50</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)  pp. 19; 50</TD></TR>
 
</table>
 
</table>
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{{TEX|done}}
  
 
[[Category:Topology]]
 
[[Category:Topology]]

Revision as of 17:04, 19 October 2014

of a normal topological space $X$

The topological invariant $\mathrm{locdim}(X)$, defined as follows. One says that $\mathrm{locdim}(X) \le n$, $n = -1,0,1,\ldots$ if for any point $x \in X$ there is a neighbourhood $O_x$ for which the Lebesgue dimension of its closure satisfies the relation $\dim \bar O_x \le n$. If $\mathrm{locdim}(X) \le n$ for some $n$, then the local dimension of $X$ is finite, so one writes $\mathrm{locdim}(X) < \infty$ and puts $$ \mathrm{locdim}(X) = \min\{ n : \mathrm{locdim}(X) \le n \} $$

Always $\mathrm{locdim}(X) \le \dim(X)$; there are normal spaces $X$ with $\mathrm{locdim}(X) < \dim(X)$; in the class of paracompact spaces always $\mathrm{locdim}(X) = \dim(X)$. If in the definition of local dimension the Lebesgue dimension $\dim \bar O_x $ is replaced by the large inductive dimension $\mathrm{Ind} \bar O_x $, then one obtains the definition of the local large inductive dimension $\mathrm{locInd}(X)$.

Comments

See [a1] for a construction of a space with $\mathrm{locdim}(X) < \dim(X)$ and — as an application — a hereditarily normal space $Y$ with $\dim Y = 0$ yet $Y$ contains subspaces of arbitrary high dimension.

For the notions of the local dimension at a point of an analytic space, algebraic variety or scheme cf. Analytic space; Dimension of an associative ring; Analytic set, and Spectrum of a ring.

References

[a1] E. Pol, R. Pol, "A hereditarily normal strongly zero-dimensional space containing subspaces of arbitrarily large dimension" Fund. Math. , 102 (1979) pp. 137–142
[a2] R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50
How to Cite This Entry:
Local dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_dimension&oldid=33942