Studying on my own currently and having trouble understanding a topology proof.
Theorem: Compact subsets of metric spaces are closed
Let be a compact subset of a metric space . We shall prove that the complement of is an open subset of
Suppose , . If , let and be neighbourhoods of and respectively, of radius less than
Since is compact, there are finitely many points in such that
If , then is a neighbourhood of which does not intersect . Hence , so that is an interior point of . The theorem follows.
I am confused as to how and are defined
Is it that
Also, how does the chosen radius play a role in the proof?