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''cellular decomposition''
 
''cellular decomposition''
  
A [[Cell complex|cell complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274701.png" /> satisfying the following conditions: (C) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274702.png" /> the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274703.png" /> is finite, that is, consists of a finite number of cells. (For any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274704.png" /> of a cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274706.png" /> is the notation for the intersection of all subcomplexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274707.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274708.png" />.) (W) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c0274709.png" /> is some subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747010.png" /> and if for any cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747012.png" /> the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747013.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747014.png" /> (and therefore in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747015.png" /> as well), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747016.png" /> is a closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747017.png" />. In this connection, each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747018.png" /> belongs to a definite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747021.png" />.
+
A [[Cell complex|cell complex]] $  X $
 +
satisfying the following conditions: (C) for any $  x \in X $
 +
the complex $  X (x) $
 +
is finite, that is, consists of a finite number of cells. (For any subset $  A $
 +
of a cell complex $  X $,  
 +
$  X (A) $
 +
is the notation for the intersection of all subcomplexes of $  X $
 +
containing $  A $.)  
 +
(W) If $  F $
 +
is some subset of $  X $
 +
and if for any cell $  t $
 +
in $  X $
 +
the intersection $  F \cap \overline{t}\; $
 +
is closed in $  \overline{t}\; $(
 +
and therefore in $  X $
 +
as well), then $  F $
 +
is a closed subset of $  X $.  
 +
In this connection, each point $  x \in X $
 +
belongs to a definite set $  t _ {x} $
 +
of $  X $,  
 +
and $  X (x) = X ( t _ {x} ) = X ( \overline{t}\; _ {x} ) $.
  
 
The notation CW comes from the initial letters of the (English) names for the above two conditions — (C) for closure finiteness and (W) for weak topology.
 
The notation CW comes from the initial letters of the (English) names for the above two conditions — (C) for closure finiteness and (W) for weak topology.
  
A finite cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747022.png" /> satisfies both conditions (C) and (W). More generally, a cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747023.png" /> each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747024.png" /> of which is contained in some finite subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747025.png" /> is a CW-complex. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747026.png" /> be a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747028.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747029.png" /> for each cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747031.png" />. Then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747032.png" /> the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747033.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747034.png" />. If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747035.png" /> does not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747036.png" />, then the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747037.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747038.png" /> and does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747039.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747040.png" /> is open and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747041.png" /> is closed.
+
A finite cell complex $  X $
 +
satisfies both conditions (C) and (W). More generally, a cell complex $  X $
 +
each point $  x $
 +
of which is contained in some finite subcomplex $  Y (x) $
 +
is a CW-complex. Let $  F $
 +
be a subset of $  X $
 +
such that $  F \cap \overline{t}\; $
 +
is closed in $  \overline{t}\; $
 +
for each cell $  t $
 +
in $  X $.  
 +
Then for any $  x \in Y $
 +
the intersection $  F \cap Y (x) $
 +
is closed in $  X $.  
 +
If the point $  x $
 +
does not belong to $  F $,  
 +
then the open set $  U _ {x} = X \setminus  ( F \cap Y (x) ) $
 +
contains $  x $
 +
and does not intersect $  F $.  
 +
The set $  ( X \setminus  F ) = \cup _ {x \in X \setminus  F }  U _ {x} $
 +
is open and $  F $
 +
is closed.
  
The class of CW-complexes (or the class of spaces of the same homotopy type as a CW-complex) is the most suitable class of topological spaces in relation to homotopy theory. Thus, if a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747042.png" /> of a CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747043.png" /> is closed, then a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747044.png" /> from the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747045.png" /> into a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747046.png" /> is continuous if and only if the restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747047.png" /> to the closures of the cells of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747048.png" /> are continuous. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747049.png" /> is a compact subset of a CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747050.png" />, then the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747051.png" /> is finite. There exists for every cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747052.png" /> of a CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747053.png" /> a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747054.png" /> that is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747055.png" /> and has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747056.png" /> as a deformation retract.
+
The class of CW-complexes (or the class of spaces of the same homotopy type as a CW-complex) is the most suitable class of topological spaces in relation to homotopy theory. Thus, if a subset $  A $
 +
of a CW-complex $  X $
 +
is closed, then a mapping $  f $
 +
from the topological space $  A $
 +
into a topological space $  Y $
 +
is continuous if and only if the restrictions of $  f $
 +
to the closures of the cells of $  X $
 +
are continuous. If $  C $
 +
is a compact subset of a CW-complex $  X $,  
 +
then the complex $  X (C) $
 +
is finite. There exists for every cell $  t $
 +
of a CW-complex $  X $
 +
a set $  D $
 +
that is open in $  \overline{t}\; $
 +
and has $  \overline{t}\; \setminus  t $
 +
as a deformation retract.
  
In practice, CW-complexes are constructed by an inductive procedure: Each stage consists in glueing cells of given dimension to the result of the previous stage. The cellular structure of such a complex is directly related to its homotopy properties. Even for such  "good"  spaces as polyhedra it is helpful to consider their representation as CW-complexes: There are usually fewer in such a representation than in a simplicial triangulation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747057.png" /> is obtained by attaching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747058.png" />-dimensional cells to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747059.png" />, then the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747061.png" />, is a strong deformation retract of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747062.png" />.
+
In practice, CW-complexes are constructed by an inductive procedure: Each stage consists in glueing cells of given dimension to the result of the previous stage. The cellular structure of such a complex is directly related to its homotopy properties. Even for such  "good"  spaces as polyhedra it is helpful to consider their representation as CW-complexes: There are usually fewer in such a representation than in a simplicial triangulation. If $  X $
 +
is obtained by attaching $  n $-
 +
dimensional cells to the space $  A $,  
 +
then the subset $  X \times 0 \cup A \times I $,  
 +
where $  I = [ 0 , 1 ] $,
 +
is a strong deformation retract of $  X \times I $.
  
A relative CW-complex is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747063.png" /> consisting of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747064.png" /> and a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747065.png" />, together with a sequence of closed subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747067.png" />, satisfying the following conditions: a) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747068.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747069.png" /> by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747070.png" />-cells; b) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747072.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747073.png" /> by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747074.png" />-dimensional cells; c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747075.png" />; d) the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747076.png" /> is compatible with the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747077.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747078.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747080.png" />-dimensional skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747081.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747082.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747083.png" />, the relative CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747084.png" /> is a CW-complex in the previous sense and its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747085.png" />-dimensional skeleton is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747086.png" />.
+
A relative CW-complex is a pair $  ( X , A ) $
 +
consisting of a topological space $  X $
 +
and a closed subset $  A $,  
 +
together with a sequence of closed subspaces $  ( X , A )  ^ {k} $,  
 +
$  k \geq  0 $,  
 +
satisfying the following conditions: a) the space $  ( X , A )  ^ {0} $
 +
is obtained from $  A $
 +
by adjoining 0 $-
 +
cells; b) for $  k \geq  1 $,
 +
$  ( X , A )  ^ {k} $
 +
is obtained from $  ( X , A )  ^ {k-1} $
 +
by adjoining $  k $-
 +
dimensional cells; c) $  X = \cup ( X , A )  ^ {k} $;  
 +
d) the topology of $  X $
 +
is compatible with the family $  \{ ( X , A )  ^ {k} \} $.  
 +
The space $  ( X , A )  ^ {k} $
 +
is called the $  k $-
 +
dimensional skeleton of $  X $
 +
relative to $  A $.  
 +
When $  A = \emptyset $,  
 +
the relative CW-complex $  ( X , \emptyset ) = X $
 +
is a CW-complex in the previous sense and its $  k $-
 +
dimensional skeleton is $  X  ^ {k} $.
  
Examples. 1) The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747087.png" /> of simplicial complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747088.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747089.png" />, defines a relative CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747091.png" />. 2) The ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747092.png" /> is a CW-complex: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747093.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747096.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747097.png" />. The sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747098.png" /> is a subcomplex of the CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c02747099.png" />. 3) If the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470100.png" /> is a relative CW-complex, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470101.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470102.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470104.png" /> is, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470105.png" />). 4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470106.png" /> is a relative CW-complex, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470107.png" /> is a CW-complex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470109.png" /> is the quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470110.png" /> obtained by identifying all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027470/c027470111.png" /> with a single point.
+
Examples. 1) The pair $  ( K , L ) $
 +
of simplicial complexes $  K , L $,  
 +
with $  L \subset  K $,  
 +
defines a relative CW-complex $  ( | K | , | L | ) $,  
 +
where $  ( | K | , | L | )  ^ {k} = ( K  ^ {k} \cup L ) $.  
 +
2) The ball $  V  ^ {n} $
 +
is a CW-complex: $  ( V  ^ {n} )  ^ {k} = p _ {0} $
 +
for $  k < n - 1 $,  
 +
$  ( V  ^ {n} )  ^ {n-1} = S  ^ {n-1} $
 +
and $  ( V  ^ {n} )  ^ {k} = V  ^ {n} $
 +
for $  k \geq  n $.  
 +
The sphere $  S  ^ {n-1} $
 +
is a subcomplex of the CW-complex $  V  ^ {n} $.  
 +
3) If the pair $  ( X , A ) $
 +
is a relative CW-complex, then so is $  ( X \times I , A \times I ) $,  
 +
and $  ( X \times I , A \times I )  ^ {k} = (( X , A )  ^ {k} \times \{ 0 , 1 \} ) \cup ( ( X , A )  ^ {k-1} \times I ) $(
 +
when $  k = 0 $,
 +
$  ( X , A )  ^ {-1} $
 +
is, by definition, $  A $).  
 +
4) If $  ( X , A ) $
 +
is a relative CW-complex, then $  X / A $
 +
is a CW-complex and $  ( X , A )  ^ {k} = ( X / A )  ^ {k} $,  
 +
where $  X / A $
 +
is the quotient space of $  X $
 +
obtained by identifying all points of $  A $
 +
with a single point.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Teleman,  "Grundzüge der Topologie und differenzierbare Mannigfaltigkeiten" , Deutsch. Verlag Wissenschaft.  (1968)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Teleman,  "Grundzüge der Topologie und differenzierbare Mannigfaltigkeiten" , Deutsch. Verlag Wissenschaft.  (1968)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1980)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 06:29, 30 May 2020


cellular decomposition

A cell complex $ X $ satisfying the following conditions: (C) for any $ x \in X $ the complex $ X (x) $ is finite, that is, consists of a finite number of cells. (For any subset $ A $ of a cell complex $ X $, $ X (A) $ is the notation for the intersection of all subcomplexes of $ X $ containing $ A $.) (W) If $ F $ is some subset of $ X $ and if for any cell $ t $ in $ X $ the intersection $ F \cap \overline{t}\; $ is closed in $ \overline{t}\; $( and therefore in $ X $ as well), then $ F $ is a closed subset of $ X $. In this connection, each point $ x \in X $ belongs to a definite set $ t _ {x} $ of $ X $, and $ X (x) = X ( t _ {x} ) = X ( \overline{t}\; _ {x} ) $.

The notation CW comes from the initial letters of the (English) names for the above two conditions — (C) for closure finiteness and (W) for weak topology.

A finite cell complex $ X $ satisfies both conditions (C) and (W). More generally, a cell complex $ X $ each point $ x $ of which is contained in some finite subcomplex $ Y (x) $ is a CW-complex. Let $ F $ be a subset of $ X $ such that $ F \cap \overline{t}\; $ is closed in $ \overline{t}\; $ for each cell $ t $ in $ X $. Then for any $ x \in Y $ the intersection $ F \cap Y (x) $ is closed in $ X $. If the point $ x $ does not belong to $ F $, then the open set $ U _ {x} = X \setminus ( F \cap Y (x) ) $ contains $ x $ and does not intersect $ F $. The set $ ( X \setminus F ) = \cup _ {x \in X \setminus F } U _ {x} $ is open and $ F $ is closed.

The class of CW-complexes (or the class of spaces of the same homotopy type as a CW-complex) is the most suitable class of topological spaces in relation to homotopy theory. Thus, if a subset $ A $ of a CW-complex $ X $ is closed, then a mapping $ f $ from the topological space $ A $ into a topological space $ Y $ is continuous if and only if the restrictions of $ f $ to the closures of the cells of $ X $ are continuous. If $ C $ is a compact subset of a CW-complex $ X $, then the complex $ X (C) $ is finite. There exists for every cell $ t $ of a CW-complex $ X $ a set $ D $ that is open in $ \overline{t}\; $ and has $ \overline{t}\; \setminus t $ as a deformation retract.

In practice, CW-complexes are constructed by an inductive procedure: Each stage consists in glueing cells of given dimension to the result of the previous stage. The cellular structure of such a complex is directly related to its homotopy properties. Even for such "good" spaces as polyhedra it is helpful to consider their representation as CW-complexes: There are usually fewer in such a representation than in a simplicial triangulation. If $ X $ is obtained by attaching $ n $- dimensional cells to the space $ A $, then the subset $ X \times 0 \cup A \times I $, where $ I = [ 0 , 1 ] $, is a strong deformation retract of $ X \times I $.

A relative CW-complex is a pair $ ( X , A ) $ consisting of a topological space $ X $ and a closed subset $ A $, together with a sequence of closed subspaces $ ( X , A ) ^ {k} $, $ k \geq 0 $, satisfying the following conditions: a) the space $ ( X , A ) ^ {0} $ is obtained from $ A $ by adjoining $ 0 $- cells; b) for $ k \geq 1 $, $ ( X , A ) ^ {k} $ is obtained from $ ( X , A ) ^ {k-1} $ by adjoining $ k $- dimensional cells; c) $ X = \cup ( X , A ) ^ {k} $; d) the topology of $ X $ is compatible with the family $ \{ ( X , A ) ^ {k} \} $. The space $ ( X , A ) ^ {k} $ is called the $ k $- dimensional skeleton of $ X $ relative to $ A $. When $ A = \emptyset $, the relative CW-complex $ ( X , \emptyset ) = X $ is a CW-complex in the previous sense and its $ k $- dimensional skeleton is $ X ^ {k} $.

Examples. 1) The pair $ ( K , L ) $ of simplicial complexes $ K , L $, with $ L \subset K $, defines a relative CW-complex $ ( | K | , | L | ) $, where $ ( | K | , | L | ) ^ {k} = ( K ^ {k} \cup L ) $. 2) The ball $ V ^ {n} $ is a CW-complex: $ ( V ^ {n} ) ^ {k} = p _ {0} $ for $ k < n - 1 $, $ ( V ^ {n} ) ^ {n-1} = S ^ {n-1} $ and $ ( V ^ {n} ) ^ {k} = V ^ {n} $ for $ k \geq n $. The sphere $ S ^ {n-1} $ is a subcomplex of the CW-complex $ V ^ {n} $. 3) If the pair $ ( X , A ) $ is a relative CW-complex, then so is $ ( X \times I , A \times I ) $, and $ ( X \times I , A \times I ) ^ {k} = (( X , A ) ^ {k} \times \{ 0 , 1 \} ) \cup ( ( X , A ) ^ {k-1} \times I ) $( when $ k = 0 $, $ ( X , A ) ^ {-1} $ is, by definition, $ A $). 4) If $ ( X , A ) $ is a relative CW-complex, then $ X / A $ is a CW-complex and $ ( X , A ) ^ {k} = ( X / A ) ^ {k} $, where $ X / A $ is the quotient space of $ X $ obtained by identifying all points of $ A $ with a single point.

References

[1] C. Teleman, "Grundzüge der Topologie und differenzierbare Mannigfaltigkeiten" , Deutsch. Verlag Wissenschaft. (1968) (Translated from Rumanian)
[2] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[3] A. Dold, "Lectures on algebraic topology" , Springer (1980)

Comments

CW-complexes have been introduced by J.H.C. Whitehead [a4] as a generalization of simplicial complexes (cf. Simplicial complex). An obvious advantage is that the number of cells needed in a decomposition is usually much smaller than the number of simplices in a triangulation. This is particularly profitable when computing homology and cohomology, and fundamental groups (cf. Fundamental group; [a1]). CW-complexes have proved useful in the context of classifying spaces for homotopy functors, and occur as Eilenberg–MacLane spaces (cf. Eilenberg–MacLane space).

Two textbooks specialized on CW-complexes are [a2] and [a3].

References

[a1] R. Brown, "Elements of modern topology" , McGraw-Hill (1968)
[a2] G.E. Cooke, P.L. Finney, "Homology of cell complexes" , Princeton Univ. Press (1967)
[a3] A.T. Lundell, S. Weingram, "The topology of CW-complexes" , v. Nostrand (1969)
[a4] J.H.C. Whitehead, "Combinatorial homotopy I" Bull. Amer. Math. Soc. , 55 (1949) pp. 213–245
How to Cite This Entry:
CW-complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CW-complex&oldid=15603
This article was adapted from an original article by D.O. Baladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article