View source for Inductive dimension

''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" />''

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.

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]]).

====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>



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

====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>