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:
“Each man has his own vocation. The talent is the call. There is one direction in which all space is open to him. He has faculties silently inviting him thither to endless exertion. He is like a ship in the river; he runs against obstructions on every side but one; on that side all obstruction is taken away, and he sweeps serenely over a deepening channel into an infinite sea.”
—Ralph Waldo Emerson (1803–1882)
“Since time immemorial, one the dry earth, scraped to the bone, of this immeasurable country, a few men travelled ceaselessly, they owned nothing, but they served no one, free and wretched lords in a strange kingdom. Janine did not know why this idea filled her with a sadness so soft and so vast that she closed her eyes. She only knew that this kingdom, which had always been promised to her would never be her, never again, except at this moment.”
—Albert Camus 1013–1960, French-Algerian novelist, dramatist, philosopher. Janine in Algeria, in The Fall, p. 27, Gallimard (9157)