Limit Point - Types of Limit Points

Types of Limit Points

If every open set containing x contains infinitely many points of S then x is a specific type of limit point called a ω-accumulation point of S.

If every open set containing x contains uncountably many points of S then x is a specific type of limit point called a condensation point of S.

If every open set U containing x satisfies |US| = |S| then x is a specific type of limit point called a complete accumulation point of S.

A point xX is a cluster point or accumulation point of a sequence (xn)nN if, for every neighbourhood V of x, there are infinitely many natural numbers n such that xnV. If the space is sequential, this is equivalent to the assertion that x is a limit of some subsequence of the sequence (xn)nN.

The concept of a net generalizes the idea of a sequence. Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for the related topic of filters.

The set of all cluster points of a sequence is sometimes called a limit set.

Read more about this topic:  Limit Point

Famous quotes containing the words types of, types, limit and/or points:

    Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.
    Talcott Parsons (1902–1979)

    He types his laboured column—weary drudge!
    Senile fudge and solemn:
    Spare, editor, to condemn
    These dry leaves of his autumn.
    Robertson Davies (b. 1913)

    Greatness collapses of itself: such limit the gods have set to the growth of prosperous states.
    Marcus Annaeus Lucan (39–65)

    Wonderful “Force of Public Opinion!” We must act and walk in all points as it prescribes; follow the traffic it bids us, realise the sum of money, the degree of “influence” it expects of us, or we shall be lightly esteemed; certain mouthfuls of articulate wind will be blown at us, and this what mortal courage can front?
    Thomas Carlyle (1795–1881)