# adjunction space

Let $X$ and $Y$ be topological spaces^{}, and let $A$ be a subspace^{} of $Y$. Given a continuous function^{} $f:A\to X,$ define the space $Z:=X{\cup}_{f}Y$ to be the quotient space^{} $X\coprod Y/\sim ,$ where the symbol $\coprod $ stands for disjoint union^{} and the equivalence relation^{} $\sim $ is generated by

$$y\sim f(y)\mathit{\hspace{1em}}\text{for all}\mathit{\hspace{1em}}y\in A.$$ |

$Z$ is called an adjunction of $Y$ to $X$ along $f$ (or along $A$, if the map $f$ is understood). This construction has the effect of gluing the subspace $A$ of $Y$ to its image in $X$ under $f.$

###### Remark 1

Though the definition makes sense for arbitrary $A$, it is usually assumed that $A$ is a closed subspace of $Y$. This results in better-behaved adjunction spaces (e.g., the quotient of $X$ by a non-closed set is never Hausdorff^{}).

###### Remark 2

The adjunction space construction is a special case of the pushout in the category of topological spaces. The two maps being pushed out are $f$ and the inclusion map^{} of $A$ into $Y$.

Title | adjunction space |
---|---|

Canonical name | AdjunctionSpace |

Date of creation | 2013-03-22 13:25:56 |

Last modified on | 2013-03-22 13:25:56 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 10 |

Author | antonio (1116) |

Entry type | Definition |

Classification | msc 54B17 |

Related topic | QuotientSpace |

Defines | adjunction |