is a covering map. is connected. Suppose such that has cardinality . Show that has cardinality .
Why does the base space need to be connected? Here is what I have:
Denote by the cardinality of .
Let . Let be an open neighborhood of evenly covered by . where the are disjoint. Then each contains a distinct element of that maps to . That is, .
Now suppose that there is some such that (is strictly greater than ). But this implies that there are at least distinct elements in the preimage of , contradicting .
That seemed like it went way too quick and way too simply relative to the standard exercises in my course. What gives?