# continuity is preserved when codomain is extended

###### Theorem 1.

Suppose $X\mathrm{,}Y$ are topological space^{} and
let $Z\mathrm{\subseteq}Y$ be equipped with the subspace topology.
If

$$f:X\to Z$$ |

is continuous, then

$$f:X\to Y$$ |

is continuous.

###### Proof.

Let $U\subseteq Y$ be an open set. Then

${f}^{-1}(U)$ | $=$ | $\mathrm{\{}x\in X:f(x)\in U\}$ | ||

$=$ | $\mathrm{\{}x\in X:f(x)\in U\cap Z\}$ | |||

$=$ | ${f}^{-1}(U\cap Z).$ |

Since $U\cap Z$ is open in $Z$, ${f}^{-1}(U)$ is open in $X$. ∎

Title | continuity is preserved when codomain is extended |
---|---|

Canonical name | ContinuityIsPreservedWhenCodomainIsExtended |

Date of creation | 2013-03-22 15:17:14 |

Last modified on | 2013-03-22 15:17:14 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 9 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 54C05 |

Related topic | IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous |