Topology

Dense-in-itself

In general topology, a subset of a topological space is said to be dense-in-itself or crowdedif has no isolated point.Equivalently, is dense-in-itself if every point of is a limit point of .Thus is dense-in-itself if and only if , where is the derived set of . A dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.) The notion of dense set is unrelated to dense-in-itself. This can sometimes be confusing, as "X is dense in X" (always true) is not the same as "X is dense-in-itself" (no isolated point). (Wikipedia).

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From playlist Physical Science Chapter 3 (Complete chapter)

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Related pages

Neighbourhood (mathematics) | Topological space | Dense set | Dense order | Counterexamples in Topology | General topology | Closure (topology) | Rational number | Glossary of topology | Perfect set | Isolated point | Nowhere dense set | Derived set (mathematics) | Singleton (mathematics) | T1 space | Closed set