View source for Movable space

A [[Compact space|compact space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202501.png" />, embedded in the [[Hilbert cube|Hilbert cube]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202502.png" />, is movable provided every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202504.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202505.png" /> admits a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202506.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202507.png" /> such that, for any other neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202508.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202509.png" />, there exists a [[Homotopy|homotopy]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025010.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025012.png" />. In other words, sufficiently small neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025013.png" /> can be deformed arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025014.png" /> [[#References|[a2]]]. K. Borsuk proved that movability is a shape invariant. The solenoids (cf. [[Solenoid|Solenoid]]) are examples of non-movable continua.

The question whether movable continua are always pointed movable is still (1998) open.

For movable spaces various shape-theoretic results assume simpler form. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025015.png" /> is a pointed shape morphism between pointed movable metric continua (cf. also [[Pointed space|Pointed space]]; [[Continuum|Continuum]]; [[Shape theory|Shape theory]]), which induces isomorphisms of the shape groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025016.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025017.png" /> and if the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025019.png" /> are finite-dimensional, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025020.png" /> is a pointed shape equivalence. This is a consequence of the shape-theoretic Whitehead theorem (cf. also [[Homotopy type|Homotopy type]]; [[Homotopy group|Homotopy group]]) and the fact that such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025021.png" /> induces isomorphisms of homotopy pro-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025022.png" /> [[#References|[a6]]], [[#References|[a5]]].

Borsuk also introduced the notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025023.png" />-movability. A compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025024.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025026.png" />-movable provided every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025029.png" /> admits a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025032.png" /> such that, for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025034.png" />, any compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025035.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025036.png" /> and any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025037.png" />, there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025040.png" /> are homotopic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025041.png" />. Clearly, if a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025042.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025043.png" />-movable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025045.png" /> is movable. Moreover, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025047.png" />-compactum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025048.png" />-movable [[#References|[a3]]]. The notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025049.png" />-movability was the beginning of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025051.png" />-shape theory, which was especially developed by A.Ch. Chigogidze [[#References|[a4]]] (cf. also [[Shape theory|Shape theory]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025052.png" />-shape theory is an important tool in the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025053.png" />-dimensional Menger manifolds, developed by M. Bestvina [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Bestvina,   "Characterizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025054.png" />-dimensional universal Menger compacta"  ''Memoirs Amer. Math. Soc.'' , '''71''' :  380  (1988)  pp. 1–110</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,   "On movable compacta"  ''Fund. Math.'' , '''66'''  (1969)  pp. 137–146</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Borsuk,   "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025055.png" />-movability"  ''Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys.'' , '''20'''  (1972)  pp. 859–864</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.Ch. Chigogidze,   "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025056.png" />-shape"  ''Uspekhi Mat. Nauk'' , '''44''' :  5  (1989)  pp. 117–140  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Dydak,   "The Whitehead and the Smale theorems in shape theory"  ''Dissert. Math.'' , '''156'''  (1979)  pp. 1–55</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.E. Keesling,   "On the Whitehead theorem in shape theory"  ''Fund. Math.'' , '''92'''  (1976)  pp. 247–253</TD></TR></table>