Perfect set
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In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S' , where S' denotes the set of all limit points of S , also known as the derived set of S . (Some authors do not consider the empty set to be perfect. ) In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of S and any neighborhood of the point, there is another point of S that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of S belongs to S . The term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being a G δ space. As another possible source of confusion, having the perfect set property is not the same as being a 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 S is perfect if S=S' , where S' denotes the set of all limit points of S , also known as the derived set of S . (Some authors do not consider the empty set to be perfect. ) In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of S and any neighborhood of the point, there is another point of S that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of S belongs to S . The term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being a G δ space. As another possible source of confusion, having the perfect set property is not the same as being a perfect set.
Examples
Examples of perfect subsets of the real line \mathbb{R} are the empty set, all closed intervals, the real line itself, and the Cantor set. The latter is noteworthy in that it is totally disconnected. Whether a set is perfect or not (and whether it is closed or not) depends on the surrounding space. For instance, the set S=[0,1]\cap \Q is perfect as a subset of the space \Q but not perfect as a subset of the space \mathbb{R} , since it fails to be closed in the latter.
See also
Dense-in-itself Finite intersection property Subspace topology
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