Namespaces
Variants
Actions

Difference between revisions of "Cell complex"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A separable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211301.png" /> that is a union of non-intersecting cells. Here, by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211303.png" />-dimensional cell one means a topological space that is homeomorphic to the interior of the unit cube of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211304.png" />. If for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211305.png" />-dimensional cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211306.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211307.png" /> one is given a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211308.png" /> from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c0211309.png" />-dimensional cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113010.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113011.png" /> such that: 1) the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113013.png" /> to the interior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113015.png" /> is one-to-one and the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113016.png" /> is the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113019.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113020.png" /> is a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113021.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113022.png" />); and 2) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113024.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113025.png" />, is contained in the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113026.png" /> of the cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113029.png" /> is called a cell complex; the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113030.png" /> is called the skeleton of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113031.png" /> of the cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113032.png" />. An example of a cell complex is a simplicial polyhedron.
+
<!--
 +
c0211301.png
 +
$#A+1 = 40 n = 0
 +
$#C+1 = 40 : ~/encyclopedia/old_files/data/C021/C.0201130 Cell complex
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113033.png" /> of a cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113034.png" /> is called a subcomplex if it is a union of cells of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113035.png" /> containing the closures of such cells. Thus, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113036.png" />-dimensional skeleton <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113038.png" /> is a subcomplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113039.png" />. Any union and any intersection of subcomplexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113040.png" /> are subcomplexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021130/c02113041.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A separable space  $  X $
 +
that is a union of non-intersecting cells. Here, by a  $  p $-
 +
dimensional cell one means a topological space that is homeomorphic to the interior of the unit cube of dimension  $  p $.  
 +
If for each  $  p $-
 +
dimensional cell  $  t  ^ {p} $
 +
of  $  X $
 +
one is given a continuous mapping  $  f $
 +
from the  $  p $-
 +
dimensional cube  $  I  ^ {p} $
 +
into  $  X $
 +
such that: 1) the restriction  $  f ^ { * } $
 +
of  $  f $
 +
to the interior  $  \mathop{\rm Int}  I  ^ {p} $
 +
of  $  I  ^ {p} $
 +
is one-to-one and the image  $  f ( I  ^ {p} ) $
 +
is the closure  $  \overline{ {t  ^ {p} }}\; $
 +
in  $  X $
 +
of  $  t  ^ {p} $(
 +
here  $  f ^ { * } $
 +
is a homeomorphism of  $  \mathop{\rm Int}  I  ^ {p} $
 +
onto  $  t  ^ {p} $);
 +
and 2) the set  $  f ( \partial  I  ^ {p} ) $,
 +
where  $  \partial  I  ^ {p} $
 +
is the boundary of  $  I  ^ {p} $,
 +
is contained in the union  $  X  ^ {p-} 1 $
 +
of the cells  $  t  ^ {p-} 1 $
 +
of  $  X $,
 +
then  $  X $
 +
is called a cell complex; the union  $  X  ^ {p-} 1 $
 +
is called the skeleton of dimension  $  p - 1 $
 +
of the cell complex  $  X $.  
 +
An example of a cell complex is a simplicial polyhedron.
 +
 
 +
A subset  $  L $
 +
of a cell complex  $  X $
 +
is called a subcomplex if it is a union of cells of $  X $
 +
containing the closures of such cells. Thus, the $  n $-
 +
dimensional skeleton $  X  ^ {n} $
 +
of $  X $
 +
is a subcomplex of $  X $.  
 +
Any union and any intersection of subcomplexes of $  X $
 +
are subcomplexes of $  X $.
  
 
Any topological space can be regarded as a cell complex — as the union of its points, which are cells of dimension 0. This example shows that the notion of a cell complex is too broad; therefore narrower classes of cell complexes are important in applications, for example the class of cellular decompositions or CW-complexes (cf. [[CW-complex|CW-complex]]).
 
Any topological space can be regarded as a cell complex — as the union of its points, which are cells of dimension 0. This example shows that the notion of a cell complex is too broad; therefore narrower classes of cell complexes are important in applications, for example the class of cellular decompositions or CW-complexes (cf. [[CW-complex|CW-complex]]).

Latest revision as of 16:43, 4 June 2020


A separable space $ X $ that is a union of non-intersecting cells. Here, by a $ p $- dimensional cell one means a topological space that is homeomorphic to the interior of the unit cube of dimension $ p $. If for each $ p $- dimensional cell $ t ^ {p} $ of $ X $ one is given a continuous mapping $ f $ from the $ p $- dimensional cube $ I ^ {p} $ into $ X $ such that: 1) the restriction $ f ^ { * } $ of $ f $ to the interior $ \mathop{\rm Int} I ^ {p} $ of $ I ^ {p} $ is one-to-one and the image $ f ( I ^ {p} ) $ is the closure $ \overline{ {t ^ {p} }}\; $ in $ X $ of $ t ^ {p} $( here $ f ^ { * } $ is a homeomorphism of $ \mathop{\rm Int} I ^ {p} $ onto $ t ^ {p} $); and 2) the set $ f ( \partial I ^ {p} ) $, where $ \partial I ^ {p} $ is the boundary of $ I ^ {p} $, is contained in the union $ X ^ {p-} 1 $ of the cells $ t ^ {p-} 1 $ of $ X $, then $ X $ is called a cell complex; the union $ X ^ {p-} 1 $ is called the skeleton of dimension $ p - 1 $ of the cell complex $ X $. An example of a cell complex is a simplicial polyhedron.

A subset $ L $ of a cell complex $ X $ is called a subcomplex if it is a union of cells of $ X $ containing the closures of such cells. Thus, the $ n $- dimensional skeleton $ X ^ {n} $ of $ X $ is a subcomplex of $ X $. Any union and any intersection of subcomplexes of $ X $ are subcomplexes of $ X $.

Any topological space can be regarded as a cell complex — as the union of its points, which are cells of dimension 0. This example shows that the notion of a cell complex is too broad; therefore narrower classes of cell complexes are important in applications, for example the class of cellular decompositions or CW-complexes (cf. CW-complex).

How to Cite This Entry:
Cell complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cell_complex&oldid=46292
This article was adapted from an original article by D.O. Baladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article