View source for CW-complex

''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" />.

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.

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.

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" />.

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" />.

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.

====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>



====Comments====
CW-complexes have been introduced by J.H.C. Whitehead [[#References|[a4]]] as a generalization of simplicial complexes (cf. [[Simplicial complex|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|Fundamental group]]; [[#References|[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|Eilenberg–MacLane space]]).

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Brown,   "Elements of modern topology" , McGraw-Hill  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.E. Cooke,   P.L. Finney,   "Homology of cell complexes" , Princeton Univ. Press  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.T. Lundell,   S. Weingram,   "The topology of CW-complexes" , v. Nostrand  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.H.C. Whitehead,   "Combinatorial homotopy I"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 213–245</TD></TR></table>