Brevity has been chosen over accuracy because the whole point is that you should know this stuff already.
Chapter 2: Basic Topology (+ some Ch. 3)
An isolated point of E is in E but not a limit point of it. E is perfect if it is exactly equal to its set of limit points. Equivalently, it is closed and has no isolated points. Ex. 2.44: The Cantor set is perfect.
A compact set is a set for which every open cover has a finite subcover.
Compactness or compact sets have these properties (with made-up names):