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/or closed:
“Don: Why are they closed? They’re all closed, every one of them.
Pawnbroker: Sure they are. It’s Yom Kippur.
Don: It’s what?
Pawnbroker: It’s Yom Kippur, a Jewish holiday.
Don: It is? So what about Kelly’s and Gallagher’s?
Pawnbroker: They’re closed, too. We’ve got an agreement. They keep closed on Yom Kippur and we don’t open on St. Patrick’s.”
—Billy Wilder (b. 1906)
“Because you live, O Christ,
the spirit bird of hope is freed for flying,
our cages of despair no longer keep us closed and life-denying.
The stone has rolled away and death cannot imprison!
O sing this Easter Day, for Jesus Christ has risen!”
—Shirley Erena Murray (20th century)