Relation To Connected Spaces
Given a topological space X, it is sometimes useful to consider whether it is possible for a subset A to be separated from its complement. This is certainly true if A is either the empty set or the entire space X, but there may be other possibilities. A topological space X is connected if these are the only two possibilities. Conversely, if a nonempty subset A is separated from its own complement, and if the only subset of A to share this property is the empty set, then A is an open-connected component of X. (In the degenerate case where X is itself the empty set {}, authorities differ on whether {} is connected and whether {} is an open-connected component of itself.)
For more on connected spaces, see Connected space.
Read more about this topic: Separated Sets
Famous quotes containing the words relation to, relation, connected and/or spaces:
“The difference between objective and subjective extension is one of relation to a context solely.”
—William James (1842–1910)
“The problem of the twentieth century is the problem of the color-line—the relation of the darker to the lighter races of men in Asia and Africa, in America and the islands of the sea. It was a phase of this problem that caused the Civil War.”
—W.E.B. (William Edward Burghardt)
“We can’t nourish our children if we don’t nourish ourselves.... Parents who manage to stay married, sane, and connected to each other share one basic characteristic: The ability to protect even small amounts of time together no matter what else is going on in their lives.”
—Ron Taffel (20th century)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)