# CW-complex

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)