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 ContinuousImagesOfPathConnectedSpacesArePathConnected 2013-03-22 15:52:38 2013-03-22 15:52:38 mps (409) mps (409) 6 mps (409) Result msc 54D05