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:
“I not deny
The jury, passing on the prisoner’s life,
May in the sworn twelve have a thief or two
Guiltier than him they try. What’s open made to justice,
That justice seizes.”
—William Shakespeare (1564–1616)
“We are closed in, and the key is turned
On our uncertainty;”
—William Butler Yeats (1865–1939)