# Movable space

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A compact space $X$, embedded in the Hilbert cube $Q$, is movable provided every neighbourhood $U$ of $X$ in $Q$ admits a neighbourhood $U ^ { \prime }$ of $X$ such that, for any other neighbourhood $U ^ { \prime \prime } \subseteq U$ of $X$, there exists a homotopy $H : U ^ { \prime } \times I \rightarrow U$ with $H _ { 0 } | _ { U ^ { \prime } } = \operatorname{id}$, $H _ { 1 } ( U ^ { \prime } ) \subseteq U ^ { \prime \prime }$. In other words, sufficiently small neighbourhoods of $X$ can be deformed arbitrarily close to $X$ [a2]. K. Borsuk proved that movability is a shape invariant. The solenoids (cf. Solenoid) are examples of non-movable continua.
For movable spaces various shape-theoretic results assume simpler form. E.g., if $f : ( X , * ) \rightarrow ( Y , * )$ is a pointed shape morphism between pointed movable metric continua (cf. also Pointed space; Continuum; Shape theory), which induces isomorphisms of the shape groups $f _ { \# } : \check{\pi} _ { k } ( X , * ) \rightarrow \check{\pi} _ { k } ( Y , * )$, for all $k$ and if the spaces $X$, $Y$ are finite-dimensional, then $f$ is a pointed shape equivalence. This is a consequence of the shape-theoretic Whitehead theorem (cf. also Homotopy type; Homotopy group) and the fact that such an $f$ induces isomorphisms of homotopy pro-groups $\pi _ { k } ( \mathcal{X} , * ) \rightarrow \pi _ { k } ( \mathcal{Y} , * )$ [a6], [a5].
Borsuk also introduced the notion of $n$-movability. A compactum $X \subseteq { Q }$ is $n$-movable provided every neighbourhood $U$ of $X$ in $Q$ admits a neighbourhood $U ^ { \prime }$ of $X$ in $Q$ such that, for any neighbourhood $U ^ { \prime \prime } \subseteq U$ of $X$, any compactum $K$ of dimension $\operatorname { dim} K \leq n$ and any mapping $f : K \rightarrow U ^ { \prime }$, there exists a mapping $g : K \rightarrow U ^ { \prime \prime }$ such that $f$ and $g$ are homotopic in $U$. Clearly, if a compactum $X$ is $n$-movable and $\operatorname{dim} X \leq n$, then $X$ is movable. Moreover, every $L C ^ { n - 1 }$-compactum is $n$-movable [a3]. The notion of $n$-movability was the beginning of the $n$-shape theory, which was especially developed by A.Ch. Chigogidze [a4] (cf. also Shape theory). The $n$-shape theory is an important tool in the theory of $n$-dimensional Menger manifolds, developed by M. Bestvina [a1].