Hilbert cube

The Hilbert cube , also called Hilbert cuboid or Hilbert fundamental cuboid , English Hilbert cube , is a topological space named after the mathematician David Hilbert , which generalizes the cube known from the visual space to an infinite number of dimensions . ${\displaystyle [0,1]^{3}}$

definition

The Hilbert cube is the product space , provided with the product topology . This means in detail: ${\displaystyle W}$ ${\displaystyle [0,1]^{\aleph _{0}}}$

• ${\displaystyle W}$is the set of all consequences with for all . ${\displaystyle x=(\xi _{n})_{n}}$ ${\displaystyle 0\leq \xi _{n}\leq 1}$ ${\displaystyle n}$
• A sequence in , where , converges to one if and only if for all indices . ${\displaystyle (x_{m})_{m}}$ ${\displaystyle W}$ ${\displaystyle x_{m}=(\xi _{n}^{(m)})_{n}}$ ${\displaystyle x=(\xi _{n})_{n}\in W}$ ${\displaystyle \lim _{m\to \infty }\xi _{n}^{(m)}=\xi _{n}}$ ${\displaystyle n\in \mathbb {N} }$

properties

• The Hilbert cube is connected and path-connected , because these properties are transferred to product spaces.
• The Hilbert cube is a compact Hausdorff space , as follows directly from Tychonoff's theorem.
• The Hilbert cube can be metrised , a metric defining the topology is through

${\displaystyle d((\xi _{n})_{n},(\eta _{n})_{n}):=\sum _{n=1}^{\infty }{\frac {|\xi _{n}-\eta _{n}|}{2^{n}}}}$

given.

• Like all compact, metrizable spaces, the Hilbert cube is separable and satisfies the second axiom of countability (and thus also the first axiom of countability ). Here is the amount

${\displaystyle D=\{(\xi _{n})_{n}\in W;\,\xi _{n}\in \mathbb {Q} {\mbox{ und }}\xi _{n}=0{\mbox{ für fast alle }}n\}}$

a countable dense subset of . The set of all -balls (with regard to the above metric) around the points from is then a countable basis. ${\displaystyle W}$ ${\displaystyle {\tfrac {1}{m}}}$ ${\displaystyle D}$

• The Lebesgue coverage dimension of the Hilbert cube is infinite, because for each the Hilbert cube contains the subspace that is too homeomorphic , so it must have a dimension for all and that means . ${\displaystyle W}$ ${\displaystyle m}$ ${\displaystyle [0,1]^{m}}$ ${\displaystyle W_{m}:=\{(\xi _{n})_{n}\in W;\,\xi _{n}=0{\mbox{ für alle }}n>m\}}$ ${\displaystyle \geq m}$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle \dim W=\infty }$

Universal property

Compact spaces with a countable base

According to the above properties, the Hilbert cube is a compact Hausdorff space with a countable basis. is universal with regard to these properties in the sense that it contains a copy of each such space. The following applies: ${\displaystyle W}$ ${\displaystyle W}$

• Every compact Hausdorff space with a countable base is homeomorphic to a closed subspace of the Hilbert cube.

Polish rooms

Also Polish spaces can be embedded in the Hilbert cube. The following applies:

• The Polish spaces are, except for homeomorphism, exactly the quantities in the Hilbert cube. ${\displaystyle G_{\delta }}$
• The compact, Polish spaces are, apart from homeomorphism, exactly the closed sets in the Hilbert cube.

The Hilbert cube in the l ²

A homeomorphic copy of the Hilbert cube can be found in the Hilbert space of the square summable sequences . Define ${\displaystyle \ell ^{2}}$

${\displaystyle {\tilde {W}}:=\{(\xi _{n})_{n}\in \ell ^{2};\,|\xi _{n}|\leq {\tfrac {1}{n}}{\mbox{ für alle }}n\}}$.

Then there is a homeomorphism if one provides with the subspace topology of the norm topology of the Hilbert space . Note that there is no null neighborhood in , because it does not contain a norm sphere. Furthermore, the relative standard topology and the relatively weak topology coincide. ${\displaystyle \textstyle \varphi \colon W\rightarrow {\tilde {W}},(\xi _{n})_{n}\mapsto ({\frac {2\xi _{n}-1}{n}})_{n}}$ ${\displaystyle {\tilde {W}}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle {\tilde {W}}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle {\tilde {W}}}$ ${\displaystyle {\tilde {W}}}$

Alternative definitions of the Hilbert cube would be or or , provided with the product topology. With such a definition itself would be a subset of the Hilbert space . The first variant is used in, where the author does not speak of the Hilbert cube because of the different side lengths, but of the Hilbert cuboid, also in, where the third variant is used for the definition. ${\displaystyle \textstyle W:=\prod _{n=1}^{\infty }[0,{\frac {1}{2^{n}}}]}$ ${\displaystyle \textstyle W:=\prod _{n=1}^{\infty }[-{\frac {1}{n}},{\frac {1}{n}}]}$ ${\displaystyle \textstyle W:=\prod _{n=1}^{\infty }[0,{\frac {1}{n}}]}$ ${\displaystyle W}$ ${\displaystyle \ell ^{2}}$

literature

• Lutz Führer: General topology with applications . Vieweg Verlag , Braunschweig 1977, ISBN 3-528-03059-3 .
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag , Stuttgart 1975, ISBN 3-519-12200-6 . MR0423277
• Stephen Willard : General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley , Reading, Massachusetts (et al.) 1970. MR0264581

Individual evidence

1. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 Chapter 5.2, sentence 8.
2. ↑ Oliver Deiser: Real Numbers. The classical continuum and the natural consequences. 2nd, corrected and enlarged edition. Springer, Berlin et al. 2008, ISBN 978-3-540-79375-5 , corollary on p. 335.
3. ^ Wolfgang Franz : Topology. Volume 1: General Topology (= Göschen Collection. Vol. 6181). 4th, improved and enlarged edition. de Gruyter, Berlin et al. 1973, ISBN 3-11-004117-0 , p. 14.
4. ^ Klaus Jänich : Topology. 8th edition. Springer, Berlin et al. 2005, ISBN 3-540-21393-7 , p. 199.