Hahn-Mazurkiewicz theorem

A nonempty Hausdorff space is the continuous image of $[0,1]$ exactly when it is compact, connected, locally connected, and second-countable.

A Peano continuum is a compact, connected, locally connected metric space. By the Hahn-Mazurkiewicz theorem, every Peano continuum is the image of a continuous surjection from $[0,1]$.