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<title>Inductive dimension</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Inductive dimension</span></span>

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<p>In the mathematical field of <a href="Topology" title="Topology">topology</a>, the <b>inductive dimension</b> of a <a href="Topological_space" title="Topological space">topological space</a> <i>X</i> is either of two values, the <b>small inductive dimension</b> ind(<i>X</i>) or the <b>large inductive dimension</b> Ind(<i>X</i>). These are based on the observation that, in <i>n</i>-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <i>R</i><sup><i>n</i></sup>, (<i>n</i> − 1)-dimensional <a href="Sphere" title="Sphere">spheres</a> (that is, the <a href="Boundary_(topology)" title="Boundary (topology)">boundaries</a> of <i>n</i>-dimensional balls) have dimension <i>n</i> − 1. Therefore it should be possible to define the dimension of a space <a href="Mathematical_induction" title="Mathematical induction">inductively</a> in terms of the dimensions of the boundaries of suitable <a href="Open_set" title="Open set">open sets</a>.

</p><p>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 <a href="Metric_space" title="Metric space">metric space</a>). The other is the <a href="Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering dimension</a>. 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.

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<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

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

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ind} (\varnothing )=\operatorname {Ind} (\varnothing )=-1}</annotation>

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</math></span><img src="./53d8b88549d1edc3c0ec6881e9a3831326584d4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.059ex; height:2.843ex;" alt="{\displaystyle \operatorname {ind} (\varnothing )=\operatorname {Ind} (\varnothing )=-1}" loading="lazy"></span></dd></dl>

<p>Then inductively, ind(<i>X</i>) is the smallest <i>n</i> such that, for every <i><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math></span><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span></i> and every open set <i>U</i> containing <i>x</i>, there is an open set <i>V</i> containing <i>x</i>, such that the <a href="Closure_(topology)" title="Closure (topology)">closure</a> of <i>V</i> is a <a href="Subset" title="Subset">subset</a> of <i>U</i>, and the boundary of <i>V</i> has small inductive dimension less than or equal to <i>n</i> − 1. (If <i>X</i> is a Euclidean <i>n</i>-dimensional space, <i>V</i> can be chosen to be an <i>n</i>-dimensional ball centered at <i>x</i>.)

</p><p>For the large inductive dimension, we restrict the choice of <i>V</i> still further; Ind(<i>X</i>) is the smallest <i>n</i> such that, for every <a href="Closed_set" title="Closed set">closed</a> subset <i>F</i> of every open subset <i>U</i> of <i>X</i>, there is an open <i>V</i> in between (that is, <i>F</i> is a subset of <i>V</i> and the closure of <i>V</i> is a subset of <i>U</i>), such that the boundary of <i>V</i> has large inductive dimension less than or equal to <i>n</i> − 1.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Relationship_between_dimensions">Relationship between dimensions</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle \dim }</annotation>

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</math></span><img src="./66115c83c4bb19068adb45849f0f596647c18f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.875ex; height:2.176ex;" alt="{\displaystyle \dim }" loading="lazy"></span> be the Lebesgue covering dimension. For any <a href="Topological_space" title="Topological space">topological space</a> <i>X</i>, we have

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<annotation encoding="application/x-tex">{\displaystyle \dim X=0}</annotation>

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</math></span><img src="./f791015b08019c26cd66f6a9d28e447637cb038c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.503ex; height:2.176ex;" alt="{\displaystyle \dim X=0}" loading="lazy"></span> if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ind} X=0.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ind} X=0.}</annotation>

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</math></span><img src="./de8f4b04ef87f93f6b145c16a6e7e875c3f1c569.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.7ex; height:2.176ex;" alt="{\displaystyle \operatorname {Ind} X=0.}" loading="lazy"></span></dd></dl>

<p><b>Urysohn's theorem</b> states that when <i>X</i> is a <a href="Normal_space" title="Normal space">normal space</a> with a <a href="Second-countable_space" title="Second-countable space">countable base</a>, then

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<annotation encoding="application/x-tex">{\displaystyle \dim X=\operatorname {Ind} X=\operatorname {ind} X.}</annotation>

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</math></span><img src="./4bfbb2c90cae957c6acf40c8ad50e8ac34898d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.477ex; height:2.176ex;" alt="{\displaystyle \dim X=\operatorname {Ind} X=\operatorname {ind} X.}" loading="lazy"></span></dd></dl>

<p>Such spaces are exactly the <a href="Separable_space" title="Separable space">separable</a> and <a href="Metrizable" class="mw-redirect" title="Metrizable">metrizable</a> <i>X</i> (see <a href="Urysohn's_metrization_theorem" class="mw-redirect" title="Urysohn's metrization theorem">Urysohn's metrization theorem</a>).

</p><p>The <b>Nöbeling–Pontryagin theorem</b> then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the <a href="Euclidean_space" title="Euclidean space">Euclidean spaces</a>, with their usual topology. The <b>Menger–Nöbeling theorem</b> (1932) states that if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is compact metric separable and of dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>, then it embeds as a subspace of Euclidean space of dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n+1}">

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<annotation encoding="application/x-tex">{\displaystyle 2n+1}</annotation>

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</math></span><img src="./8ca410f731fe4c7c444330343afb1d1850eadaea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 2n+1}" loading="lazy"></span>. (<a href="Georg_N%C3%B6beling" title="Georg Nöbeling">Georg Nöbeling</a> was a student of <a href="Karl_Menger" title="Karl Menger">Karl Menger</a>. He introduced <b>Nöbeling space</b>, the subspace of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{2n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{2n+1}}</annotation>

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</math></span><img src="./3b5e8ef45a42d0f2453469e3ef7f87148fa14bda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.144ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{2n+1}}" loading="lazy"></span> consisting of points with at least <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

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</math></span><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"></span> co-ordinates being <a href="Irrational_number" title="Irrational number">irrational numbers</a>, which has universal properties for embedding spaces of dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</p><p>Assuming only <i>X</i> metrizable we have (<a href="Miroslav_Kat%C4%9Btov" title="Miroslav Katětov">Miroslav Katětov</a>)

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<p>or assuming <i>X</i> <a href="Compact_space" title="Compact space">compact</a> and <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> (<a href="P._S._Aleksandrov" class="mw-redirect" title="P. S. Aleksandrov">P. S. Aleksandrov</a>)

</p>

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

</p><p>A separable metric space <i>X</i> satisfies the inequality <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ind} X\leq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ind} X\leq n}</annotation>

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</math></span><img src="./a65603c166c4af7e71a7ef70162c48391a9d5a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.285ex; height:2.343ex;" alt="{\displaystyle \operatorname {Ind} X\leq n}" loading="lazy"></span> if and only if for every closed sub-space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> of the space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and each continuous mapping <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to S^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f:A\to S^{n}}</annotation>

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</math></span><img src="./8394ee6815e09a6c5d3458f26c26156df1d2f02e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.313ex; height:2.676ex;" alt="{\displaystyle f:A\to S^{n}}" loading="lazy"></span> there exists a continuous extension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {f}}:X\to S^{n}}">

<mover>

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<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {f}}:X\to S^{n}}</annotation>

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</math></span><img src="./3b06ca97b5fda02318e7302e1e1043d5fc7d12ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.97ex; height:3.009ex;" alt="{\displaystyle {\bar {f}}:X\to S^{n}}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFArkhangelskiiPontryagin1990" class="citation book cs1">Arkhangelskii, A.V.; Pontryagin, L.S. (1990). <i>General Topology</i>. Vol. I. Berlin, DE: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-18178-4</bdi>.</cite> <i>Page 104</i></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in <a href="Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, I.</a>, ed., <i>Landmark Writings in Western Mathematics</i>. Elsevier: 844-55.</li>

<li>R. Engelking, <i>Theory of Dimensions. Finite and Infinite</i>, Heldermann Verlag (1995), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-88538-010-2</bdi>.</li>

<li>V. V. Fedorchuk, <i>The Fundamentals of Dimension Theory</i>, appearing in <i>Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I</i>, (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-18178-4</bdi>.</li>

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

<li>A. R. Pears, <i>Dimension theory of general spaces</i>, Cambridge University Press (1975).</li></ul>

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</style><div id="Dimension155" style="font-size:114%;margin:0 4em"><a href="Dimension" title="Dimension">Dimension</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensional spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dimension_(vector_space)" title="Dimension (vector space)">Vector space</a></li>

<li><a href="Euclidean_space" title="Euclidean space">Euclidean space</a></li>

<li><a href="Affine_space" title="Affine space">Affine space</a></li>

<li><a href="Projective_space" title="Projective space">Projective space</a></li>

<li><a href="Free_module" title="Free module">Free module</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">Algebraic variety</a></li>

<li><a href="Spacetime" title="Spacetime">Spacetime</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other dimensions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Krull_dimension" title="Krull dimension">Krull</a></li>

<li><a href="Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering</a></li>

<li><a href="Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li>

<li><a href="Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">Minkowski</a></li>

<li><a href="Fractal_dimension" title="Fractal dimension">Fractal</a></li>

<li><a href="Degrees_of_freedom" title="Degrees of freedom">Degrees of freedom</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polytope" title="Polytope">Polytopes</a> and <a href="Shape" title="Shape">shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hyperplane" title="Hyperplane">Hyperplane</a></li>

<li><a href="Hypersurface" title="Hypersurface">Hypersurface</a></li>

<li><a href="Hypercube" title="Hypercube">Hypercube</a></li>

<li><a href="Hyperrectangle" title="Hyperrectangle">Hyperrectangle</a></li>

<li><a href="Demihypercube" title="Demihypercube">Demihypercube</a></li>

<li><a href="N-sphere" title="N-sphere">Hypersphere</a></li>

<li><a href="Cross-polytope" title="Cross-polytope">Cross-polytope</a></li>

<li><a href="Simplex" title="Simplex">Simplex</a></li>

<li><a href="Hyperpyramid" title="Hyperpyramid">Hyperpyramid</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Number systems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hypercomplex_number" title="Hypercomplex number">Hypercomplex numbers</a></li>

<li><a href="Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensions by number</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Three-dimensional_space" title="Three-dimensional space">Three</a></li>

<li><a href="Seven-dimensional_space" title="Seven-dimensional space">Seven</a></li>

<li><a href="Eight-dimensional_space" title="Eight-dimensional space">Eight</a></li>

<li><a href="Dimension" title="Dimension"><i>n</i>-dimensions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hyperspace" title="Hyperspace">Hyperspace</a></li>

<li><a href="Codimension" title="Codimension">Codimension</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b>Category</b></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2023-11-27" href="https://en.wikipedia.org/wiki/?title=Inductive_dimension&oldid=1187179338">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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