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\begin{document}

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\section*{Borsuk&#39;s shape theory}

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

\noindent\hyperlink{reprise_of_the_idea}{Reprise of the Idea}\dotfill \pageref*{reprise_of_the_idea} \linebreak
\noindent\hyperlink{some_details}{Some details}\dotfill \pageref*{some_details} \linebreak
\noindent\hyperlink{chapman_complement_theorem}{Chapman complement theorem}\dotfill \pageref*{chapman_complement_theorem} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{reprise_of_the_idea}{}\subsection*{{Reprise of the Idea}}\label{reprise_of_the_idea}

The basic idea behind Borsuk's shape theory is explained in the entry on [[shape theory]], so will not be repeated here, except to say that it considers compact metric spaces embedded in the [[Hilbert cube]], then uses the open neighbourhoods of the space as a `net' of approximations of the space. The space is, of course, the intersection of all these open neighbourhoods.

Any compact metric space can be embedded in the [[Hilbert cube]], so it is sufficient to consider just compact subspaces of that space.

\hypertarget{some_details}{}\subsection*{{Some details}}\label{some_details}

Let $s= \prod_{n=1}^\infty (-\frac{1}{n},\frac{1}{n})$ be the \emph{pseudo-interior} of the [[Hilbert cube]], $Q= \prod_{n=1}^\infty [-\frac{1}{n},\frac{1}{n}]$.

We will define (a category equivalent to) the \emph{Borsuk Shape category}, $Shape_B$, to have compact subsets of $s$ as objects and some morphisms that need a bit of explaining.

If $X$ and $Y$ are compact subsets of $s$, then a \emph{fundamental sequence}, $\underline{f} : X\to Y$, is defined to be a sequence of maps $f_n : Q\to Q$ such that for every neighbourhood $V$ of $Y$ in $Q$, there exists a neighbourhood $U$ of $X$ in $Q$ and an integer $n_0$ such that if $n, n^\prime \geq n_0$, the restrictions $f_n|_U$ and $f_{n^\prime}|_U$ are homotopic within $V$.

Note that the $f_n(X)$ do not have to be contained in $Y$, they only have to be `near' $Y$.

Two fundamental sequences, $\underline{f},\underline{f}^\prime : X\to Y$, are said to be \emph{homotopic}, $\underline{f}\sim \underline{f}^\prime$ provided that for every neighbourhood $V$ of $Y$ in $Q$, there is a neighbourhood $U$ of $X$ in $Q$ and an integer $n_0$ such that if $n \geq n_0$, then $f_n|_U$ and $f^\prime_{n}|_U$ are homotopic within $V$.

The morphisms of $Shape_B$ and taken to be the homotopy classes of fundamental sequences between the corresponding spaces.

Two compacta contained in $s$ are said to have the \emph{same shape} if they are isomorphic in $Shape_B$. As an example, the [[Warsaw circle]] has the same shape as the circle.

\hypertarget{chapman_complement_theorem}{}\subsection*{{Chapman complement theorem}}\label{chapman_complement_theorem}

\begin{utheorem}
If $X$ and $Y$ are compacta in $s$, then $X$ and $Y$ have the same shape if and only if their complements $Q\setminus X$ and $Q\setminus Y$ are homeomorphic.

\end{utheorem}
Chapman extended the association $X$ `goes to' $Q\setminus X$ to a functor from the Borsuk shape category to the weak [[proper homotopy theory|proper homotopy]] category of complements in $Q$ of compacta. This was the basis for Edwards-Hastings formulation of [[strong shape theory]], on replacing the weak form of proper homotopy by a strong form.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[shape theory]]

\begin{itemize}%
\item [[strong shape theory]]

\end{itemize}


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

\begin{itemize}%
\item K. Borsuk, \emph{Concerning homotopy properties of compacta}, Fund Math. 62 (1968) 223-254
\item K. Borsuk, \emph{Theory of Shape}, Monografie Matematyczne Tom 59,Warszawa 1975.
\item T. A.Chapman, \emph{On Some Applications of Infinite Dimensional Manifolds to the Theory of Shape}, Fund. Math. 6 (1972), 181 - 193.
\item D. A. Edwards and [[H. M. Hastings]], (1976), ech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag. [[!redirects Borsuk shape theory]]

\end{itemize}


\end{document}