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| In the mathematical field of [[topology]], the '''inductive dimension''' of a [[topological space]] ''X'' is either of two values, the '''small inductive dimension''' ind(''X'') or the '''large inductive dimension''' Ind(''X''). These are based on the observation that, in ''n''-dimensional [[Euclidean space]] ''R''<sup>''n''</sup>, (''n'' − 1)-dimensional [[sphere]]s (that is, the [[boundary (topology)|boundaries]] of ''n''-dimensional balls) have dimension ''n'' − 1. Therefore it should be possible to define the dimension of a space [[mathematical induction|inductively]] in terms of the dimensions of the boundaries of suitable [[open set]]s.
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| The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a [[metric space]]). The other is the [[Lebesgue covering dimension]]. The term "topological dimension" is ordinarily understood to refer to Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.
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| ==Formal definition==
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| We want the dimension of a point to be 0, and a point has empty boundary, so we start with
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| :<math>\operatorname{ind}(\varnothing)=\operatorname{Ind}(\varnothing)=-1</math>
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| Then inductively, ind(''X'') is the smallest ''n'' such that, for every ''<math>x \isin X</math>'' and every open set ''U'' containing ''x'', there is an open ''V'' containing ''x'', where the [[Closure (topology)|closure]] of ''V'' is a [[subset]] of ''U'', such that the boundary of ''V'' has small inductive dimension less than or equal to ''n'' − 1. (In the case above, where ''X'' is Euclidean ''n''-dimensional space, ''V'' will be chosen to be an ''n''-dimensional ball centered at ''x''.)
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| For the large inductive dimension, we restrict the choice of ''V'' still further; Ind(''X'') is the smallest ''n'' such that, for every [[closed set|closed]] subset ''F'' of every open subset ''U'' of ''X'', there is an open ''V'' in between (that is, ''F'' is a subset of ''V'' and the closure of ''V'' is a subset of ''U''), such that the boundary of ''V'' has large inductive dimension less than or equal to ''n'' − 1.
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| ==Relationship between dimensions==
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| Let <math>\operatorname{dim}</math> be the Lebesgue covering dimension. For any [[topological space]] ''X'', we have
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| :<math>\operatorname{dim} X = 0</math> if and only if <math>\operatorname{Ind} X = 0.</math> | |
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| '''Urysohn's theorem''' states that when ''X'' is a [[normal space]] with a [[second-countable space|countable base]], then
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| :<math>\operatorname{dim} X = \operatorname{Ind} X = \operatorname{ind} X</math>.
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| Such spaces are exactly the [[separable space|separable]] and [[metrizable]] ''X'' (see [[Urysohn's metrization theorem]]).
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| The '''Nöbeling-Pontryagin theorem''' then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the [[Euclidean space]]s, with their usual topology. The '''Menger-Nöbeling theorem''' (1932) states that if ''X'' is compact metric separable and of dimension ''n'', then it embeds as a subspace of Euclidean space of dimension 2''n'' + 1. ([[Georg Nöbeling]] was a student of [[Karl Menger]]. He introduced '''Nöbeling space''', the subspace of '''R'''<sup>2''n'' + 1</sup> consisting of points with at least ''n'' + 1 co-ordinates being [[irrational number]]s, which has universal properties for embedding spaces of dimension ''n''.)
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| Assuming only ''X'' metrizable we have ([[Miroslav Katětov]])
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| :ind ''X'' ≤ Ind ''X'' = dim ''X'';
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| or assuming ''X'' [[compact space|compact]] and [[Hausdorff space|Hausdorff]] ([[P. S. Aleksandrov]])
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| :dim ''X'' ≤ ind ''X'' ≤ Ind ''X''.
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| Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ.
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| A separable metric space ''X'' satisfies the inequality <math>\operatorname{Ind}X\le n</math> if and only if for every closed sub-space <math>A</math> of the space <math>X</math> and each continuous mapping <math>f:A\to S^n</math> there exists a continuous extension <math>\bar f:X\to S^n</math>.
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| ==References==
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| {{Empty section|date=July 2010}}
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| ==Further reading==
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| *Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in [[Ivor Grattan-Guinness|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. Elsevier: 844-55.
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| *R. Engelking, ''Theory of Dimensions. Finite and Infinite'', Heldermann Verlag (1995), ISBN 3-88538-010-2.
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| *V. V. Fedorchuk, ''The Fundamentals of Dimension Theory'', appearing in ''Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I'', (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.
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| *V. V. Filippov, ''On the inductive dimension of the product of bicompacta'', Soviet. Math. Dokl., 13 (1972), N° 1, 250-254.
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| *A. R. Pears, ''Dimension theory of general spaces'', Cambridge University Press (1975).
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| [[Category:Dimension theory]]
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