I am having some difficulty in understanding the difference between Perfect and Compact sets. More specifically, my problem is rather understanding how Perfect sets are different from Compact sets, by that I mean, I understand Compact Sets more than Perfect Sets.
I know that for any set, $S$, to be Compact, every sequence of $S$ has a subsequence that converges to a point which also lies in $S$. This is basic definition but is not difference than saying it is Bounded and Closed or Heine Borel Theorem.
Now, the definition of Perfect Set is $P$ is a Perfect Set if $P =P'$ where $P'$ is the set of Limit Points of $P$ (WolframAlpha). At other places, I also that a set is Perfect Set if $P$ is closed and accumulation point of $P$. Though, I do not understand this completely, it sounds similar to definition of Compact Sets.
I would appreciate any explanation.
Further, are there are any non-singleton sets that are Compact but Not Perfect? How about Perfect but Non Compact Sets?