Inductive dimension

<h2 id="formal-definition">Formal definition</h2>
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 n such that, for every 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 and every open set U containing x, there is an open set V containing x, such that the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of V is a <a href="/facts/Subset/SJGERmbA">subset</a> of U, and the boundary of V has small inductive dimension less than or equal to n − 1. (If X is a Euclidean n-dimensional space, V can be chosen to be an n-dimensional ball centered at x.)
For the large inductive dimension, we restrict the choice of V still further; Ind(X) is the smallest n such that, for every <a href="/facts/Closed_set/upYnRTUj">closed</a> 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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h2 id="relationship-between-dimensions">Relationship between dimensions</h2>
Let 
 
 
 
 dim
 
 
 {\displaystyle \dim }
 
 be the Lebesgue covering dimension. For any <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 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 <a href="/facts/Normal_space/t4DLqLCd">normal space</a> with a <a href="/facts/Second-countable_space/qw9YdnBN">countable base</a>, then

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

Such spaces are exactly the <a href="/facts/Separable_space/2bmU73bk">separable</a> and <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a> X (see <a href="/facts/Urysohn%27s_metrization_theorem/BCLdo5Jf">Urysohn's metrization theorem</a>).
The Nöbeling–Pontryagin theorem then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean spaces</a>, 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}
 
. (<a href="/facts/Georg_N%C3%B6beling/DI5PuHsP">Georg Nöbeling</a> was a student of <a href="/facts/Karl_Menger/xgI0uzL1">Karl Menger</a>. 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 <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a>, which has universal properties for embedding spaces of dimension 
 
 
 
 n
 
 
 {\displaystyle n}
 
.)
Assuming only X metrizable we have (<a href="/facts/Miroslav_Kat%C4%9Btov/WjRSWlDD">Miroslav Katětov</a>)

ind X ≤ Ind X = dim X;
or assuming X <a href="/facts/Compact_space/d0cgXJH7">compact</a> and <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> (<a href="/facts/P._S._Aleksandrov/booz6U09">P. S. Aleksandrov</a>)

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}}
 
.

<h2 id="further-reading">Further reading</h2>
<ul><li>Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in <a href="/facts/Ivor_Grattan-Guinness/j8LPQtrS">Grattan-Guinness, I.</a>, ed., Landmark Writings in Western Mathematics. Elsevier: 844-55.</li>
<li>R. Engelking, Theory of Dimensions. Finite and Infinite, Heldermann Verlag (1995), <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-88538-010-2.</li>
<li>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 <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-18178-4.</li>
<li>V. V. Filippov, On the inductive dimension of the product of bicompacta, Soviet. Math. Dokl., 13 (1972), N° 1, 250-254.</li>
<li>A. R. Pears, Dimension theory of general spaces, Cambridge University Press (1975).</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Arkhangelskii, A.V.; Pontryagin, L.S. (1990). General Topology. Vol. I. Berlin, DE: Springer-Verlag. ISBN 3-540-18178-4. Page 104 <a href="3-540-18178-4" target="_blank">3-540-18178-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Inductive dimension open-in-new

Inductive dimension