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 |U ∩ S| = |S| then x is a specific type of limit point called a complete accumulation point of S.
A point x ∈ X is a cluster point or accumulation point of a sequence (xn)n ∈ N if, for every neighbourhood V of x, there are infinitely many natural numbers n such that xn ∈ V. If the space is sequential, this is equivalent to the assertion that x is a limit of some subsequence of the sequence (xn)n ∈ N.
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:
“The wider the range of possibilities we offer children, the more intense will be their motivations and the richer their experiences. We must widen the range of topics and goals, the types of situations we offer and their degree of structure, the kinds and combinations of resources and materials, and the possible interactions with things, peers, and adults.”
—Loris Malaguzzi (1920–1994)
“Our children evaluate themselves based on the opinions we have of them. When we use harsh words, biting comments, and a sarcastic tone of voice, we plant the seeds of self-doubt in their developing minds.... Children who receive a steady diet of these types of messages end up feeling powerless, inadequate, and unimportant. They start to believe that they are bad, and that they can never do enough.”
—Stephanie Martson (20th century)
“We are rarely able to interact only with folks like ourselves, who think as we do. No matter how much some of us deny this reality and long for the safety and familiarity of sameness, inclusive ways of knowing and living offer us the only true way to emancipate ourselves from the divisions that limit our minds and imaginations.”
—bell hooks (b. 1955)
“We only part to meet again.
Change, as ye list, ye winds: my heart shall be
The faithful compass that still points to thee.”
—John Gay (1685–1732)