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# cell attachment

Let $X$ be a topological space, and let $Y$ be the adjunction $Y:=X\cup_{\varphi}D^{k}$, where $D^{k}$ is a closed $k$-ball and $\varphi\colon S^{{k-1}}\rightarrow X$ 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 $\varphi.$ The image $e^{k}$ of $D^{k}$ in $Y$ is called a closed $k$-cell, and the image $\smash{\overset{\circ}{e}}^{k}$ 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 to the 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 attachment if $Y$ is homeomorphic to an adjunction $X\cup_{{\left\{\varphi_{i}\right\}}}D^{{k_{i}}}$, where the maps ${\left\{\varphi_{i}\right\}}$ into $X$ are defined on the boundary spheres of closed balls ${\left\{D^{{k_{i}}}\right\}}$.

## Mathematics Subject Classification

54B15*no label found*

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