View source for Local dimension

''of a normal topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060120/l0601201.png" />''

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

<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>

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====
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.

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]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Pol,   R. Pol,   "A hereditarily normal strongly zero-dimensional space containing subspaces of arbitrarily large dimension"  ''Fund. Math.'' , '''102'''  (1979)  pp. 137–142</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>

[[Category:Topology]]