Perfect set

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To understand what a perfect set is, one must understand the concept of a limit point (aka accumulation point). A point p is a limit point of a set S if for any sized neighborhood around p the neighborhood contains at least one point of S other than p. In contrast, an isolated point is a point p for which there exists a neighborhood around p that contains no other points of S.

The derived set of a set S (usually denoted S^') is the set of all limit points of S. A set S is said to be a perfect set if it is S = S^'. Equivalently, S is perfect if it is closed and has no isolated points.

Examples of Perfect Sets