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| In [[mathematics]], the '''Hilbert cube''', named after [[David Hilbert]], is a [[topological space]] that provides an instructive example of some ideas in [[topology]]. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).
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| ==Definition==
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| The Hilbert cube is best defined as the [[topological product]] of the [[interval (mathematics)|intervals]] [0, 1/''n''] for ''n'' = 1, 2, 3, 4, ... That is, it is a [[cuboid]] of [[countably infinite]] [[dimension]], where the lengths of the edges in each orthogonal direction form the sequence <math>\lbrace 1/n \rbrace_{n\in\mathbb{N}}</math>.
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| The Hilbert cube is [[homeomorphism|homeomorphic]] to the product of [[countably infinite]]ly many copies of the [[unit interval]] [0, 1]. In other words, it is topologically indistinguishable from the [[unit cube]] of countably infinite dimension.
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| If a point in the Hilbert cube is specified by a sequence <math>\lbrace a_n \rbrace</math> with <math>0 \leq a_n \leq 1/n</math>, then a homeomorphism to the infinite dimensional unit cube is given by <math>h : a_n \rarr n\cdot a_n</math>.
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| ==The Hilbert cube as a metric space==
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| It is sometimes convenient to think of the Hilbert cube as a [[metric space]], indeed as a specific subset of a separable [[Hilbert space]] (i.e. a Hilbert space with a countably infinite Hilbert basis).
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| For these purposes, it is best not to think of it as a product of copies of [0,1], but instead as
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| :[0,1] × [0,1/2] × [0,1/3] × ···;
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| as stated above, for topological properties, this makes no difference.
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| That is, an element of the Hilbert cube is an [[infinite sequence]]
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| :(''x''<sub>''n''</sub>)
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| that satisfies
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| :0 ≤ ''x''<sub>''n''</sub> ≤ 1/''n''.
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| Any such sequence belongs to the Hilbert space [[Lp_space#The_p-norm_in_countably_infinite_dimensions|ℓ<sub>2</sub>]], so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the [[product topology]] in the above definition.
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| ==Properties==
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| As a product of [[compact (topology)|compact]] [[Hausdorff space]]s, the Hilbert cube is itself a compact Hausdorff space as a result of the [[Tychonoff theorem]].
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| The compactness of the Hilbert cube can also be proved without the Axiom of Choice by constructing a continuous function from the usual [[Cantor set]] onto the Hilbert cube.
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| In ℓ<sub>2</sub>, no point has a compact [[neighbourhood (topology)|neighbourhood]] (thus, ℓ<sub>2</sub> is not [[locally compact]]). One might expect that all of the compact subsets of ℓ<sub>2</sub> are finite-dimensional.
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| The Hilbert cube shows that this is not the case.
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| But the Hilbert cube fails to be a neighbourhood of any point ''p'' because its side becomes smaller and smaller in each dimension, so that an [[open ball]] around ''p'' of any fixed radius ''e'' > 0 must go outside the cube in some dimension.
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| Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore [[Normal space|T4]]) and [[second countable]]. It is more interesting that the converse also holds: Every [[second countable]] [[Normal space|T4]] space is homeomorphic to a subset of the Hilbert cube.
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| Every G<sub>δ</sub>-subset of the Hilbert cube is a [[Polish space]], a topological space homeomorphic to a separable and complete metric space. Conversely, every Polish space is homeomorphic to a [[Gδ set|G<sub>δ</sub>-subset]] of the Hilbert cube.<ref>[[#Srivastava|Srivastava]], pp. 55</ref>
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| == Notes ==
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| {{reflist|2}}
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| == References ==
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| * <cite id="Srivastava">{{Cite book | last = Srivastava | first = Sashi Mohan | authorlink = | title = A Course on Borel Sets | url = http://books.google.com/?id=FhYGYJtMwcUC | accessdate = 12-04-08 | publisher = [[Springer-Verlag]] | series = Graduate Texts in Mathematics | year = 1998 | doi = | isbn = 978-0-387-98412-4 }}</cite>
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| * {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=507446}} | year=1995 | postscript=<!--None-->}}
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| [[Category:Topological spaces]]
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