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\begin{document}

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\section*{Hilbert cube}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{pseudointerior_of_}{Pseudo-interior of $Q$}\dotfill \pageref*{pseudointerior_of_} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The Hilbert cube is the product:

\begin{displaymath}
\prod_n [0,\frac{1}{n}]\cong\prod_n [-\frac{1}{n},\frac{1}{n}]
\end{displaymath}
It is a [[compactum|compact]] [[metrizable space]] under the sup norm, where the metric topology equals the product topology (as is easily seen). It is variously denoted by $Q$ or $I^\omega$.

It plays a central role in [[Borsuk's shape theory]], and is the basis for the construction of Hilbert cube manifolds. The theory of these were developed by Tom Chapman (mid 1970s) and were used in his proof of the topological invariance of [[Whitehead torsion]].

\hypertarget{pseudointerior_of_}{}\subsection*{{Pseudo-interior of $Q$}}\label{pseudointerior_of_}

It has an important subspace known as its \emph{pseudo-interior}. This is the product of the corresponding open intervals,

\begin{displaymath}
s= \prod_n (-\frac{1}{n},\frac{1}{n}).
\end{displaymath}
This plays an essential role in the [[shape theory|Chapman complement theorem]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Let $Q$ be the Hilbert cube.

\begin{itemize}%
\item A space is [[second countable space|second countable]] (has a countable [[basis]]) and $T_4$ (is [[separation axioms|normal and Hausdorff]]) if and only if it is homeomorphic to a [[subspace]] of $Q$.


\item A [[topological space]] is [[Polish space|Polish]] if and only if it is homeomorphic to a $G_\delta$-[[G-delta subset|subset]] of $Q$.


\item The Hilbert cube has some counterintuitive properties, such as the fact that it is a [[homogeneous space]] (i.e., the group of self-homeomorphisms $Aut(Q)$ acts transitively on $Q$), even though it seems to have a ``boundary''. See \hyperlink{HW}{Halverson and Wright} for some explicit constructions.



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item T.A.Chapman, \emph{On Some Applications of Infinite Dimensional Manifolds to the Theory of Shape}, Fund. Math. \&6 (1972), 181 - 193.


\item T.A. Chapman, \emph{Lectures on Hilbert Cube Manifolds}, CBMS 28, American Mathematical Society, Providence, RI, 1975


\item Denise M. Halverson, David G. Wright, \emph{The Homogeneous Property of the Hilbert Cube}, \href{http://arxiv.org/pdf/1211.1363v1.pdf}{http://arxiv.org/pdf/1211.1363v1.pdf}



\end{itemize}
[[!redirects Hilbert cube]] [[!redirects Hilbert Cube]]



\end{document}