Open and Closed Sets, Topology and Convergence
Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.
About any point in a metric space we define the open ball of radius about as the set
These open balls form the base for a topology on M, making it a topological space.
Explicitly, a subset of is called open if for every in there exists an such that is contained in . The complement of an open set is called closed. A neighborhood of the point is any subset of that contains an open ball about as a subset.
A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details.
A sequence in a metric space is said to converge to the limit iff for every, there exists a natural number N such that for all . Equivalently, one can use the general definition of convergence available in all topological spaces.
A subset of the metric space is closed iff every sequence in that converges to a limit in has its limit in .
Read more about this topic: Metric Space
Famous quotes containing the words open and, open and/or closed:
“I think, for the rest of my life, I shall refrain from looking up things. It is the most ravenous time-snatcher I know. You pull one book from the shelf, which carries a hint or a reference that sends you posthaste to another book, and that to successive others. It is incredible, the number of books you hopefully open and disappointedly close, only to take down another with the same result.”
—Carolyn Wells (1862–1942)
“I’ve tried to open the door. My knock isn’t that big a sound. But it is like the knock in “The Wizard of Oz.” It set up this echo through the halls until it was heard by everyone.”
—Shannon Faulkner (b. c. 1975)
“No other creative field is as closed to those who are not white and male as is the visual arts. After I decided to be an artist, the first thing that I had to believe was that I, a black woman, could penetrate the art scene, and that, further, I could do so without sacrificing one iota of my blackness or my femaleness or my humanity.”
—Faith Ringgold (b. 1934)