# continuous images of path connected spaces are path connected

###### Proposition.

The continuous^{} image of a path connected space is path connected.

###### Proof.

Let X be a path connected space, and suppose f is a
continuous surjection whose domain is X. Let a and b
be points in the image of f. Each has at least one preimage^{} in
X, and by the path connectedness of X, there is a path in
X from a preimage of a to a preimage of b. Applying
f to this path yields a path in the image of f from a
to b.
∎

Title | continuous images of path connected spaces are path connected |
---|---|

Canonical name | ContinuousImagesOfPathConnectedSpacesArePathConnected |

Date of creation | 2013-03-22 15:52:38 |

Last modified on | 2013-03-22 15:52:38 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 6 |

Author | mps (409) |

Entry type | Result |

Classification | msc 54D05 |