Finite union of compact sets is compact
I think i have the proof, however I am worried about some assertions.
Pf: Let where is compact
Let be an open cover of
is also an open cover for
Since is compact , a finite subcover
is a finite subcover for
I am not sure if I can just say that an open cover of K exists. It makes sense, to me anyway, that I could cover any set with a bunch of open sets. Still, I am not sure if an open cover of K exists.
And the union of the finite subcovers remains finite?
Thanks for helping.