Derived set (mathematics)

Among the a lot of derivative is understood in mathematics, the set of all limit points of this set. Provided here is that on the set of a distance term, or more generally, a topology is defined. An equally important expression is the Derived quantity. Does the amount, so there are signs for their derivation, or, for the first derivative.

Higher amount of derivatives

Higher amount leads are inductively defined: the -th derivative is the derivative of the - th derivative. The completed envelope is of referred to as the zero derivative of. More generally, the -th derivative is by and for each limit ordinal by defined for each isolated ordinal.

Properties

The derivative of a quantity can be empty. In a T1 - space, the following rules apply:

A set is perfect if and only if. The insichdichte core of a set is the average of its derivatives.

Spaces with a countable base

Be the set of points of condensation. In a topological space with a countable base applies:

• First Set of Lindelöf:
• Set of Cantor - Bendixson, I: Every closed set can be represented as the union of a perfect and an at most countable set. In Polish rooms, this representation is unique.

This leads to the conclusion:

• Every closed set is either countable or has the cardinality of the continuum.

One possible use of evidence

• Set of Cantor - Bendixson, II: In spaces with a countable base always ends for each subset of the result of their derivatives with a perfect amount, ie for any amount exists an ordinal such that.

The smallest such ordinal is called Cantor Bendixsonscher degree of quantity.

The second set of Cantor - Bendixson is a generalization of the first. Consider the induced on M by X topology. If the Cantor Benidixsonsche degree of quantity is in this room, then

The quantities \ consist only of isolated points and are at most countable. The amount

Is at most countable as a union of at most countably many at most countable sets itself. The amount due is perfect.