cell attachment

Let $X$ be a topological space,and let $Y$ be the adjunction$Y:=X{\cup }_{\phi }{D}^k$,where $D^k$ is a closed $k$-ball (http://planetmath.org/StandardNBall)and $\phi :S^{k-1}\to X$ is a continuous map,with $S^{k-1}$ is the $(k-1)$-sphere considered as the boundary of $D^k$.Then, we say that $Y$ is obtained from $X$by the attachment of a $k$-cell, by the attaching map $\phi .$The image $e^k$ of $D^k$ in $Y$ is called a closed $k$-cell,and the image ${\stackrel{\circ }{e}}^k$ of the interior

$$D^{\circ }:=D^k\setminus S^{k-1}$$

of $D^k$ is the corresponding open $k$-cell.

Note that for $k=0$ the above definition reduces tothe statement that $Y$ is the disjoint union of $X$ with a one-point space.

More generally, we say that $Y$ is obtained from $X$ by cell attachmentif $Y$ is homeomorphic to an adjunction $X{\cup }_{\left\{{\phi }_i\right\}}{D}^{k_i}$,where the maps $\left\{{\phi }_i\right\}$ into $X$are defined on the boundary spheres of closed balls $\left\{D^{k_i}\right\}$.