i think i'm just having trouble visualizing this problem...
they define x in S as an isolated point of S if there exists r>0 such that B(x;r)∩S={x}. the problem asks to show that the closure of subset S of X is the disjoint union of limit pts of S and isolated pts of S.
how would i go about solving this problem?
i think disjoint union just means a collection. i just looked at the solutions in the back and it seems to suggest a proof by contradiction, i.e. assume that x is not a limit point of S and to mess around with an epsilon ball containing only finitely many pts of S. i was just wondering what the best way to approach this problem was. thanks.
It should be clear from the definition that no isolated point is a limit point of S.
It follows that every non-isolated point is a limit point of S.
Now let denote the set of all limits points of S.
Then would be the set of all isolated points of S.
Observing that the closure of S consists of all points of S union all limits points of S, the proof follows.