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<title>Zero-dimensional space</title>
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</style><div role="note" class="hatnote navigation-not-searchable">This article is about zero dimension in topology. For several kinds of zero space in algebra, see <a href="Zero_object_(algebra)" title="Zero object (algebra)">zero object (algebra)</a>.</div>
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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><div class="sidebar-caption"><a href="Projective_geometry" title="Projective geometry">Projecting</a> a <a href="Sphere" title="Sphere">sphere</a> to a <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li>
<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a>
<ul><li><a href="Elliptic_geometry" title="Elliptic geometry">Elliptic</a>
<ul><li><a href="Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li>
<li><a href="Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li>
<li><a href="Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li>
<li><a href="Projective_geometry" title="Projective geometry">Projective</a></li>
<li><a href="Affine_geometry" title="Affine geometry">Affine</a></li>
<li><a href="Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li>
<li><a href="Analytic_geometry" title="Analytic geometry">Analytic</a></li>
<li><a href="Algebraic_geometry" title="Algebraic geometry">Algebraic</a>
<ul><li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li>
<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li>
<li><a href="Differential_geometry" title="Differential geometry">Differential</a>
<ul><li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li>
<li><a href="Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li>
<li><a href="Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li>
<li><a href="Complex_geometry" title="Complex geometry">Complex</a></li>
<li><a href="Finite_geometry" title="Finite geometry">Finite</a></li>
<li><a href="Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a>
<ul><li><a href="Digital_geometry" title="Digital geometry">Digital</a></li></ul></li>
<li><a href="Convex_geometry" title="Convex geometry">Convex</a></li>
<li><a href="Computational_geometry" title="Computational geometry">Computational</a></li>
<li><a href="Fractal" title="Fractal">Fractal</a></li>
<li><a href="Incidence_geometry" title="Incidence geometry">Incidence </a></li>
<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a>
<ul><li><a href="Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a>
<ul><li><a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul>
<ul><li><a href="Angle" title="Angle">Angle</a></li>
<li><a href="Curve" title="Curve">Curve</a></li>
<li><a href="Diagonal" title="Diagonal">Diagonal</a></li>
<li><a href="Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="Perpendicular" title="Perpendicular">Perpendicular</a>)</li>
<li><a href="Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li>
<li><a href="Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul>
<ul><li><a href="Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li>
<li><a href="Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li>
<li><a href="Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td>
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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Line_(geometry)" title="Line (geometry)">Line</a>
<ul><li><a href="Line_segment" title="Line segment">segment</a></li>
<li><a href="Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li>
<li><a href="Length" title="Length">Length</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">
<ul><li><a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li>
<li><a href="Area" title="Area">Area</a></li>
<li><a href="Polygon" title="Polygon">Polygon</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li>
<li><a href="Hypotenuse" title="Hypotenuse">Hypotenuse</a></li>
<li><a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Square" title="Square">Square</a></li>
<li><a href="Rectangle" title="Rectangle">Rectangle</a></li>
<li><a href="Rhombus" title="Rhombus">Rhombus</a></li>
<li><a href="Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Trapezoid" title="Trapezoid">Trapezoid</a></li>
<li><a href="Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Diameter" title="Diameter">Diameter</a></li>
<li><a href="Circumference" title="Circumference">Circumference</a></li>
<li><a href="Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Volume" title="Volume">Volume</a></li></ul>
<ul><li><a href="Cube" title="Cube">Cube</a>
<ul><li><a href="Cuboid" title="Cuboid">cuboid</a></li></ul></li>
<li><a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li>
<li><a href="Dodecahedron" title="Dodecahedron">Dodecahedron</a></li>
<li><a href="Icosahedron" title="Icosahedron">Icosahedron</a></li>
<li><a href="Octahedron" title="Octahedron">Octahedron</a></li>
<li><a href="Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li>
<li><a href="Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li>
<li><a href="Sphere" title="Sphere">Sphere</a></li>
<li><a href="Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Four-dimensional_space" title="Four-dimensional space">Four</a>-/other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Tesseract" title="Tesseract">Tesseract</a></li>
<li><a href="Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td>
</tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;">
<a href="List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li>
<li><a href="Aryabhata" title="Aryabhata">Aryabhata</a></li>
<li><a href="Ahmes" title="Ahmes">Ahmes</a></li>
<li><a href="Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li>
<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li>
<li><a href="Archimedes" title="Archimedes">Archimedes</a></li>
<li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>
<li><a href="Baudhayana" title="Baudhayana">Baudhayana</a></li>
<li><a href="J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li>
<li><a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a></li>
<li><a href="%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li>
<li><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li>
<li><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li>
<li><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li>
<li><a href="Euclid" title="Euclid">Euclid</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>
<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>
<li><a href="Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li>
<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>
<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>
<li><a href="Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li>
<li><a href="K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li>
<li><a href="Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li>
<li><a href="Felix_Klein" title="Felix Klein">Klein</a></li>
<li><a href="Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li>
<li><a href="Manava" title="Manava">Manava</a></li>
<li><a href="Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li>
<li><a href="Minggatu" title="Minggatu">Minggatu</a></li>
<li><a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a></li>
<li><a href="Pythagoras" title="Pythagoras">Pythagoras</a></li>
<li><a href="Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li>
<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>
<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li>
<li><a href="Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li>
<li><a href="Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li>
<li><a href="Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li>
<li><a href="Oswald_Veblen" title="Oswald Veblen">Veblen</a></li>
<li><a href="Virasena" title="Virasena">Virasena</a></li>
<li><a href="Yang_Hui" title="Yang Hui">Yang Hui</a></li>
<li><a href="Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li>
<li><a href="Zhang_Heng" title="Zhang Heng">Zhang</a></li>
<li><a href="List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">
<a href="Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Ahmes" title="Ahmes">Ahmes</a></li>
<li><a href="Baudhayana" title="Baudhayana">Baudhayana</a></li>
<li><a href="Manava" title="Manava">Manava</a></li>
<li><a href="Pythagoras" title="Pythagoras">Pythagoras</a></li>
<li><a href="Euclid" title="Euclid">Euclid</a></li>
<li><a href="Archimedes" title="Archimedes">Archimedes</a></li>
<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1–1400s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Zhang_Heng" title="Zhang Heng">Zhang</a></li>
<li><a href="K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li>
<li><a href="Aryabhata" title="Aryabhata">Aryabhata</a></li>
<li><a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a></li>
<li><a href="Virasena" title="Virasena">Virasena</a></li>
<li><a href="Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li>
<li><a href="Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li>
<li><a href="Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li>
<li><a href="Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li>
<li><a href="Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li>
<li><a href="Yang_Hui" title="Yang Hui">Yang Hui</a></li>
<li><a href="Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1400s–1700s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li>
<li><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li>
<li><a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a></li>
<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>
<li><a href="Minggatu" title="Minggatu">Minggatu</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>
<li><a href="Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li>
<li><a href="Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1700s–1900s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>
<li><a href="Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li>
<li><a href="J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li>
<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li>
<li><a href="Felix_Klein" title="Felix Klein">Klein</a></li>
<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>
<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>
<li><a href="Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li>
<li><a href="%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li>
<li><a href="Oswald_Veblen" title="Oswald Veblen">Veblen</a></li>
<li><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li>
<li><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Present day</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>
<li><a href="Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td>
</tr></tbody></table></div></div></td>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>zero-dimensional topological space</b> (or <b>nildimensional space</b>) is a <a href="Topological_space" title="Topological space">topological space</a> that has dimension zero with respect to one of several inequivalent notions of assigning a <a href="Dimension" title="Dimension">dimension</a> to a given topological space.<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> A graphical illustration of a zero-dimensional space is a <a href="Point_(geometry)" title="Point (geometry)">point</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Specifically:
</p>
<ul><li>A topological space is zero-dimensional with respect to the <a href="Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering dimension</a> if every <a href="Open_cover" class="mw-redirect" title="Open cover">open cover</a> of the space has a <a href="Refinement_(topology)" class="mw-redirect" title="Refinement (topology)">refinement</a> that is a cover by disjoint open sets.</li>
<li>A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.</li>
<li>A topological space is zero-dimensional with respect to the <a href="Small_inductive_dimension" class="mw-redirect" title="Small inductive dimension">small inductive dimension</a> if it has a <a href="Base_(topology)" title="Base (topology)">base</a> consisting of <a href="Clopen_set" title="Clopen set">clopen sets</a>.</li></ul>
<p>The three notions above agree for <a href="Separable_space" title="Separable space">separable</a>, <a href="Metrisable_space" class="mw-redirect" title="Metrisable space">metrisable spaces</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Properties_of_spaces_with_small_inductive_dimension_zero">Properties of spaces with small inductive dimension zero</h2></div>
<ul><li>A zero-dimensional <a href="Hausdorff_space" title="Hausdorff space">Hausdorff space</a> is necessarily <a href="Totally_disconnected" class="mw-redirect" title="Totally disconnected">totally disconnected</a>, but the converse fails. However, a <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (<a href="#CITEREFArhangel'skiiTkachenko2008">Arhangel'skii & Tkachenko 2008</a>, Proposition 3.1.7, p.136) for the non-trivial direction.)</li>
<li>Zero-dimensional <a href="Polish_space" title="Polish space">Polish spaces</a> are a particularly convenient setting for <a href="Descriptive_set_theory" title="Descriptive set theory">descriptive set theory</a>. Examples of such spaces include the <a href="Cantor_space" title="Cantor space">Cantor space</a> and <a href="Baire_space_(set_theory)" title="Baire space (set theory)">Baire space</a>.</li>
<li>Hausdorff zero-dimensional spaces are precisely the <a href="Subspace_topology" title="Subspace topology">subspaces</a> of topological <a href="Power_set" title="Power set">powers</a> <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 2^{I}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{I}}</annotation>
</semantics>
</math></span><img src="./4714005ae9e8fe65f1fa43660de05edd2dd172b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.223ex; height:2.676ex;" alt="{\displaystyle 2^{I}}" loading="lazy"></span> where <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 2=\{0,1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2=\{0,1\}}</annotation>
</semantics>
</math></span><img src="./dfe9491853a6dbf61328b21d83141dc74c48e72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.945ex; height:2.843ex;" alt="{\displaystyle 2=\{0,1\}}" loading="lazy"></span> is given the <a href="Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a>. Such a space is sometimes called a <a href="Cantor_cube" title="Cantor cube">Cantor cube</a>. If <span class="texhtml mvar" style="font-style:italic;">I</span> is <a href="Countable_set" title="Countable set">countably infinite</a>, <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 2^{I}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{I}}</annotation>
</semantics>
</math></span><img src="./4714005ae9e8fe65f1fa43660de05edd2dd172b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.223ex; height:2.676ex;" alt="{\displaystyle 2^{I}}" loading="lazy"></span> is the Cantor space.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Manifolds">Manifolds</h2></div>
<p>All points of a zero-dimensional <a href="Manifold" title="Manifold">manifold</a> are <a href="Isolated_point" title="Isolated point">isolated</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFArhangel'skiiTkachenko2008" class="citation book cs1"><a href="Alexander_Arhangelskii" title="Alexander Arhangelskii">Arhangel'skii, Alexander</a>; Tkachenko, Mikhail (2008). <i>Topological Groups and Related Structures</i>. Atlantis Studies in Mathematics. Vol. 1. Atlantis Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-90-78677-06-2</bdi>.</cite></li>
<li><cite id="CITEREFEngelking,_Ryszard1977" class="citation book cs1"><a href="Ryszard_Engelking" title="Ryszard Engelking">Engelking, Ryszard</a> (1977). <i>General Topology</i>. PWN, Warsaw.</cite></li>
<li><cite id="CITEREFWillard,_Stephen2004" class="citation book cs1">Willard, Stephen (2004). <i>General Topology</i>. Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-43479-6</bdi>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFHazewinkel1989" class="citation book cs1">Hazewinkel, Michiel (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8aHsCAAAQBAJ&q=zero-dimensional+space+math&pg=PA190"><i>Encyclopaedia of Mathematics, Volume 3</i></a>. Kluwer Academic Publishers. p. 190. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789400959941</bdi>.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFWolcottMcTernan2012" class="citation conference cs1">Wolcott, Luke; McTernan, Elizabeth (2012). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150626111631/http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf">"Imagining Negative-Dimensional Space"</a> <span class="cs1-format">(PDF)</span>. In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza (eds.). <i>Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture</i>. Phoenix, Arizona, USA: Tessellations Publishing. pp. <span class="nowrap">637–</span>642. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-938664-00-7</bdi>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1099-6702">1099-6702</a>. Archived from <a rel="nofollow" class="external text" href="http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 26 June 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">10 July</span> 2015</span>.</cite></span>
</li>
</ol></div></div>
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<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="Inductive_dimension" title="Inductive dimension">Inductive</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">
<ul>
<li><a href="One-dimensional_space" title="One-dimensional space">One</a></li>
<li><a href="Two-dimensional_space" title="Two-dimensional space">Two</a></li>
<li><a href="Three-dimensional_space" title="Three-dimensional space">Three</a></li>
<li><a href="Four-dimensional_space" title="Four-dimensional space">Four</a></li>
<li><a href="Five-dimensional_space" title="Five-dimensional space">Five</a></li>
<li><a href="Six-dimensional_space" title="Six-dimensional space">Six</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><!--htdig_noindex--><div><div class="zim-footer">
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