Topology | Properties of topological spaces

Perfect set

In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also known as the derived set of . In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of and any neighborhood of the point, there is another point of that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of belongs to . Note that the term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being a Gδ space. As another possible source of confusion, also note that having the perfect set property is not the same as being a perfect set. (Wikipedia).

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Related pages

Cantor space | Topological space | Countable set | Separable space | Isolated point | Continuous function | Cardinality of the continuum | Empty set | Derived set (mathematics) | Injective function | Hausdorff space | Neighbourhood (mathematics) | Descriptive set theory | Dense-in-itself | Perfect set property | General topology | Polish space | Subspace topology | Scattered space | Cardinality | Complete metric space | Cantor set | Closed set | Finite intersection property