Definition:Perfect Set
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Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.
A perfect set is a subset $S \subseteq X$ such that:
- $S = S'$
where $S'$ is the derived set of $S$.
That is, where:
- every point of $S$ is a limit point of $S$ and
- every limit point of $S$ is a point of $S$.
Alternative definitions:
- A perfect set is a closed set which has no isolated points.
- A perfect set is a set $S$ which is dense-in-itself and which contains all its limit points.
These definitions are logically equivalent.
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Limit Points
