# If $fmM\text{nonscript}:X\to Y$ is continuous then $fmM\text{nonscript}:X\to f(X)$ is continuous

###### Theorem 1.

Suppose $X\mathrm{,}Y$ are topological spaces^{} and $f\mathrm{:}X\mathrm{\to}Y$ is a
continuous function^{}. Then
$f\mathrm{:}X\mathrm{\to}f\mathit{}\mathrm{(}X\mathrm{)}$ is continuous when $f\mathit{}\mathrm{(}X\mathrm{)}$ is equipped with
the subspace topology.

###### Proof.

Let us first note that using a property on this page (http://planetmath.org/InverseImage), we have

$$X={f}^{-1}f(X).$$ |

For the proof, suppose that $A$ is open in $f(X)$, that is, $A=U\cap f(X)$ for some open set $U\subset Y$. From the properties of the inverse image, we have

$${f}^{-1}(A)={f}^{-1}(U)\cap {f}^{-1}(f(X))={f}^{-1}(U)$$ |

so ${f}^{-1}(A)$ is open in $X$. ∎

Title | If $fmM\text{nonscript}:X\to Y$ is continuous then $fmM\text{nonscript}:X\to f(X)$ is continuous |
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Canonical name | IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous |

Date of creation | 2013-03-22 15:16:28 |

Last modified on | 2013-03-22 15:16:28 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 6 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 26A15 |

Classification | msc 54C05 |

Related topic | ContinuityIsPreservedWhenCodomainIsExtended |