universal nets in compact spaces are convergent
Theorem - A universal net in a compact space is convergent.
Proof : Suppose by contradiction that was not convergent. Then for every we would find neighborhoods
such that
The collection of all this neighborhoods cover , and as is compact
, a finite number also cover .
The net is not eventually in so it must be eventually in (because it is a net). Therefore we can find such that
Because we have a finite number we can find such that for each .
Then for all , i.e. for all . But cover and thus we have a contradiction.