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 acontinuous surjection whose domain is X. Let a and bbe points in the image of f. Each has at least one preimage inX, and by the path connectedness of X, there is a path inX from a preimage of a to a preimage of b. Applyingf to this path yields a path in the image of f from ato b.∎