# 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 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 $X{\cup}_{\left\{{\phi}_{i}\right\}}{D}^{{k}_{i}}$,
where the maps $\left\{{\phi}_{i}\right\}$ into $X$
are defined on the boundary spheres of closed balls^{} $\left\{{D}^{{k}_{i}}\right\}$.

Title | cell attachment |

Canonical name | CellAttachment |

Date of creation | 2013-03-22 13:25:53 |

Last modified on | 2013-03-22 13:25:53 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 13 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54B15 |

Synonym | cell adjunction |

Related topic | CWComplex |

Defines | cell |

Defines | open cell |

Defines | closed cell |

Defines | attaching map |