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:
“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)
“If there is nothing new on the earth, still the traveler always has a resource in the skies. They are constantly turning a new page to view. The wind sets the types on this blue ground, and the inquiring may always read a new truth there.”
—Henry David Thoreau (1817–1862)
“... there are two types of happiness and I have chosen that of the murderers. For I am happy. There was a time when I thought I had reached the limit of distress. Beyond that limit, there is a sterile and magnificent happiness.”
—Albert Camus (1913–1960)
“The two great points of difference between a democracy and a republic are: first, the delegation of the government, in the latter, to a small number of citizens elected by the rest; secondly, the greater number of citizens and greater sphere of country over which the latter may be extended.”
—James Madison (1751–1836)