# Inductive dimension

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Invariant of topological spaces

In the mathematical field of [topology](/source/Topology), the **inductive dimension** of a [topological space](/source/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](/source/Euclidean_space) **R***n*, the [boundaries](/source/Boundary_(topology)) of [balls](/source/Ball_(mathematics)) have dimension *n* − 1. Therefore it should be possible to define the dimension of a general space [inductively](/source/Mathematical_induction) in terms of the dimensions of the boundaries of suitable [open sets](/source/Open_set) in that space.

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](/source/Metric_space)). The other is the [Lebesgue covering dimension](/source/Lebesgue_covering_dimension). The term "topological dimension" is ordinarily understood to refer to the Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.

## Formal definitions

We want the dimension of a point to be 0, and a point has empty boundary, so we start with

- ind ⁡ ( ∅ ) = Ind ⁡ ( ∅ ) = − 1. {\displaystyle \operatorname {ind} (\varnothing )=\operatorname {Ind} (\varnothing )=-1.}

Then inductively, ind(*X*) is the smallest natural number *n* with the following property: for every *x ∈ X {\displaystyle x\in X}* and every open set *U* containing *x*, there is an open set *V* that contains *x* and whose [closure](/source/Closure_(topology)) is contained in *U*, such that the [boundary](/source/Boundary_(topology)) of *V* has small inductive dimension less than *n*. Here, the boundary of *V* is considered as a topological space using the [subspace topology](/source/Subspace_topology) inherited from *X.* (In the case of subspaces of Euclidean space, we may think of the sets *V* as tiny balls centered at *x*.) If no such *n* exists, we write ind(*X*) = ∞.

The large inductive dimension Ind(*X*) is defined to be the smallest *n* such that, for every [closed](/source/Closed_set) subset *F* and every open subset *U* containing *F*, there is an open *V* that contains *F* and whose closure is contained in *U*, such that the boundary of *V* has large inductive dimension less than *n*. If no such *n* exists, we write Ind(*X*) = ∞.[1]

## Examples

For nice and tame spaces, the inductive dimensions yield the expected answer. Consider for instance the set

- X = { ( x , y , 0 ) ∣ x 2 + y 2 ≤ 1 } ∪ { ( 0 , 0 , z ) ∣ 0 ≤ z < 1 } ∪ { ( 0 , 0 , − 1 ) } {\displaystyle X=\{(x,y,0)\mid x^{2}+y^{2}\leq 1\}\cup \{(0,0,z)\mid 0\leq z<1\}\cup \{(0,0,-1)\}}

with the topology inherited from Euclidean space **R**3. Intuitively, *X* consists of a 2-dimensional piece attached to a 1-dimensional piece, together with a disjoint 0-dimensional point. Both large and small inductive dimensions of *X* turn out to be 2.

Maybe less expected is ind ⁡ Q = Ind ⁡ Q = 0. {\displaystyle \operatorname {ind} \mathbb {Q} =\operatorname {Ind} \mathbb {Q} =0.} This holds because for [irrational numbers](/source/Irrational_number) *a* and *b*, the set { r ∈ Q ∣ a < r < b } {\displaystyle \{r\in \mathbb {Q} \mid a<r<b\}} is both open and closed in Q {\displaystyle \mathbb {Q} } and therefore has empty boundary.

## Relationships between dimensions

Let dim {\displaystyle \dim } be the Lebesgue covering dimension. For any [topological space](/source/Topological_space) *X*, we have

- dim ⁡ X = 0 {\displaystyle \dim X=0} if and only if Ind ⁡ X = 0. {\displaystyle \operatorname {Ind} X=0.}

**Urysohn's theorem** states that when *X* is a [normal space](/source/Normal_space) with a [countable base](/source/Second-countable_space), then

- dim ⁡ X = Ind ⁡ X = ind ⁡ X . {\displaystyle \dim X=\operatorname {Ind} X=\operatorname {ind} X.}

Such spaces are exactly the [separable](/source/Separable_space) and [metrizable](/source/Metrizable) spaces (see [Urysohn's metrization theorem](/source/Urysohn's_metrization_theorem)).

The **Nöbeling–Pontryagin theorem** then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the [Euclidean spaces](/source/Euclidean_space), with their usual topology. The **Menger–Nöbeling theorem** (1932) states that if X {\displaystyle X} is compact metric separable and of dimension n {\displaystyle n} , then it embeds as a subspace of Euclidean space of dimension 2 n + 1 {\displaystyle 2n+1} . ([Georg Nöbeling](/source/Georg_N%C3%B6beling) was a student of [Karl Menger](/source/Karl_Menger). He introduced **Nöbeling space**, the subspace of R 2 n + 1 {\displaystyle \mathbf {R} ^{2n+1}} consisting of points with at least n + 1 {\displaystyle n+1} co-ordinates being [irrational numbers](/source/Irrational_number), which has universal properties for embedding spaces of dimension n {\displaystyle n} .)

Assuming only *X* metrizable we have ([Miroslav Katětov](/source/Miroslav_Kat%C4%9Btov))

- ind *X* ≤ Ind *X* = dim *X*;

or assuming *X* [compact](/source/Compact_space) and [Hausdorff](/source/Hausdorff_space) ([P. S. Aleksandrov](/source/P._S._Aleksandrov))

- dim *X* ≤ ind *X* ≤ Ind *X*.

Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ.

A separable metric space *X* satisfies the inequality Ind ⁡ X ≤ n {\displaystyle \operatorname {Ind} X\leq n} if and only if for every closed sub-space A {\displaystyle A} of the space X {\displaystyle X} and each continuous mapping f : A → S n {\displaystyle f:A\to S^{n}} there exists a continuous extension f ¯ : X → S n {\displaystyle {\bar {f}}:X\to S^{n}} .

## References

1. **[^](#cite_ref-1)** Arkhangelskii, A.V.; Pontryagin, L.S. (1990). *General Topology*. Vol. I. Berlin, DE: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [3-540-18178-4](https://en.wikipedia.org/wiki/Special:BookSources/3-540-18178-4). *Page 104*

## Further reading

- Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in [Grattan-Guinness, I.](/source/Ivor_Grattan-Guinness), ed., *Landmark Writings in Western Mathematics*. Elsevier: 844-55.

- R. Engelking, *Theory of Dimensions. Finite and Infinite*, Heldermann Verlag (1995), [ISBN](/source/ISBN_(identifier)) [3-88538-010-2](https://en.wikipedia.org/wiki/Special:BookSources/3-88538-010-2).

- V. V. Fedorchuk, *The Fundamentals of Dimension Theory*, appearing in *Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I*, (1993) [A. V. Arkhangel'skii](/source/Alexander_Arhangelskii) and [L. S. Pontryagin](/source/L._S._Pontryagin) (Eds.), Springer-Verlag, Berlin [ISBN](/source/ISBN_(identifier)) [3-540-18178-4](https://en.wikipedia.org/wiki/Special:BookSources/3-540-18178-4).

- V. V. Filippov, *On the inductive dimension of the product of bicompacta*, Soviet. Math. Dokl., 13 (1972), N° 1, 250-254.

- A. R. Pears, *Dimension theory of general spaces*, Cambridge University Press (1975).

v t e Dimension Dimensional spaces Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime Other dimensions Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category

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