In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces.
Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept.
Read more about Separated Sets: Definitions, Relation To Separation Axioms and Separated Spaces, Relation To Connected Spaces, Relation To Topologically Distinguishable Points
Famous quotes containing the words separated and/or sets:
“The doctrine of those who have denied that certainty could be attained at all, has some agreement with my way of proceeding at the first setting out; but they end in being infinitely separated and opposed. For the holders of that doctrine assert simply that nothing can be known; I also assert that not much can be known in nature by the way which is now in use. But then they go on to destroy the authority of the senses and understanding; whereas I proceed to devise helps for the same.”
—Francis Bacon (1560–1626)
“In the beautiful, man sets himself up as the standard of perfection; in select cases he worships himself in it.... Man believes that the world itself is filled with beauty—he forgets that it is he who has created it. He alone has bestowed beauty upon the world—alas! only a very human, an all too human, beauty.”
—Friedrich Nietzsche (1844–1900)