We prove a lemma that the covering map is an open map.
Let be open in . Let , then has an evenly covered open neighborhood , such that , where the are disjoint open sets in , and is a homeomorphism. is open in , and open in , so is open in , thus open in .
There exists such that . Thus so for some . Thus and thus . This shows is an interior point of . Hence is open, thus is an open map.