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**cribby** It looks like I ended up with something similar to your first suggestion. I essentially showed that for any two points in an evenly covered open neighborhood having k-cardinality preimage, the preimage of the two points must have cardinality k. Then used this with a set construction similar to your suggestion (sorry, I'm being code-lazy this morning), showed this set must be open and its complement must be open. The connectedness of the base space establishes that the constructed set is, in fact, the entire base space as it is nonempty by assumption.

If its not too much of a bother, however, I would appreciate a little more enlightenment on why connectedness is necessary. Perhaps a counterexample to the statement with a non-connected base space?

Thank you much!